Laplace transform - Wikipedia

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class="vector-toc-list"> <li id="toc-Bilateral_Laplace_transform" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bilateral_Laplace_transform"> <div class="vector-toc-text"> 2.1 Bilateral Laplace transform </div> </a> <ul id="toc-Bilateral_Laplace_transform-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inverse_Laplace_transform" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inverse_Laplace_transform"> <div class="vector-toc-text"> 2.2 Inverse Laplace transform </div> </a> <ul id="toc-Inverse_Laplace_transform-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probability_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probability_theory"> <div class="vector-toc-text"> 2.3 Probability theory </div> </a> <ul id="toc-Probability_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_construction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_construction"> <div class="vector-toc-text"> 2.4 Algebraic construction </div> </a> <ul id="toc-Algebraic_construction-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Region_of_convergence" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Region_of_convergence"> <div class="vector-toc-text"> 3 Region of convergence </div> </a> <ul id="toc-Region_of_convergence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_and_theorems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties_and_theorems"> <div class="vector-toc-text"> 4 Properties and theorems </div> </a> <button aria-controls="toc-Properties_and_theorems-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties and theorems subsection </button> <ul id="toc-Properties_and_theorems-sublist" class="vector-toc-list"> <li id="toc-Relation_to_power_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_power_series"> <div class="vector-toc-text"> 4.1 Relation to power series </div> </a> <ul id="toc-Relation_to_power_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_moments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_moments"> <div class="vector-toc-text"> 4.2 Relation to moments </div> </a> <ul id="toc-Relation_to_moments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transform_of_a_function's_derivative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transform_of_a_function's_derivative"> <div class="vector-toc-text"> 4.3 Transform of a function's derivative </div> </a> <ul id="toc-Transform_of_a_function's_derivative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Evaluating_integrals_over_the_positive_real_axis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Evaluating_integrals_over_the_positive_real_axis"> <div class="vector-toc-text"> 4.4 Evaluating integrals over the positive real axis </div> </a> <ul id="toc-Evaluating_integrals_over_the_positive_real_axis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relationship_to_other_transforms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relationship_to_other_transforms"> <div class="vector-toc-text"> 5 Relationship to other transforms </div> </a> <button aria-controls="toc-Relationship_to_other_transforms-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Relationship to other transforms subsection </button> <ul id="toc-Relationship_to_other_transforms-sublist" class="vector-toc-list"> <li id="toc-Laplace–Stieltjes_transform" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Laplace–Stieltjes_transform"> <div class="vector-toc-text"> 5.1 Laplace–Stieltjes transform </div> </a> <ul id="toc-Laplace–Stieltjes_transform-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fourier_transform" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fourier_transform"> <div class="vector-toc-text"> 5.2 Fourier transform </div> </a> <ul id="toc-Fourier_transform-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mellin_transform" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mellin_transform"> <div class="vector-toc-text"> 5.3 Mellin transform </div> </a> <ul id="toc-Mellin_transform-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Z-transform" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Z-transform"> <div class="vector-toc-text"> 5.4 Z-transform </div> </a> <ul id="toc-Z-transform-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Borel_transform" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Borel_transform"> <div class="vector-toc-text"> 5.5 Borel transform </div> </a> <ul id="toc-Borel_transform-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fundamental_relationships" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fundamental_relationships"> <div class="vector-toc-text"> 5.6 Fundamental relationships </div> </a> <ul id="toc-Fundamental_relationships-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Table_of_selected_Laplace_transforms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Table_of_selected_Laplace_transforms"> <div class="vector-toc-text"> 6 Table of selected Laplace transforms </div> </a> <ul id="toc-Table_of_selected_Laplace_transforms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-s-domain_equivalent_circuits_and_impedances" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#s-domain_equivalent_circuits_and_impedances"> <div class="vector-toc-text"> 7 s-domain equivalent circuits and impedances </div> </a> <ul id="toc-s-domain_equivalent_circuits_and_impedances-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples_and_applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Examples_and_applications"> <div class="vector-toc-text"> 8 Examples and applications </div> </a> <button aria-controls="toc-Examples_and_applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Examples and applications subsection </button> <ul id="toc-Examples_and_applications-sublist" class="vector-toc-list"> <li id="toc-Evaluating_improper_integrals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Evaluating_improper_integrals"> <div class="vector-toc-text"> 8.1 Evaluating improper integrals </div> </a> <ul id="toc-Evaluating_improper_integrals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_impedance_of_a_capacitor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_impedance_of_a_capacitor"> <div class="vector-toc-text"> 8.2 Complex impedance of a capacitor </div> </a> <ul id="toc-Complex_impedance_of_a_capacitor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Impulse_response" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Impulse_response"> <div class="vector-toc-text"> 8.3 Impulse response </div> </a> <ul id="toc-Impulse_response-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Phase_delay" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Phase_delay"> <div class="vector-toc-text"> 8.4 Phase delay </div> </a> <ul id="toc-Phase_delay-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Statistical_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Statistical_mechanics"> <div class="vector-toc-text"> 8.5 Statistical mechanics </div> </a> <ul id="toc-Statistical_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spatial_(not_time)_structure_from_astronomical_spectrum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spatial_(not_time)_structure_from_astronomical_spectrum"> <div class="vector-toc-text"> 8.6 Spatial (not time) structure from astronomical spectrum </div> </a> <ul id="toc-Spatial_(not_time)_structure_from_astronomical_spectrum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Birth_and_death_processes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Birth_and_death_processes"> <div class="vector-toc-text"> 8.7 Birth and death processes </div> </a> <ul id="toc-Birth_and_death_processes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tauberian_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tauberian_theory"> <div class="vector-toc-text"> 8.8 Tauberian theory </div> </a> <ul id="toc-Tauberian_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 9 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 10 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 11 References </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle References subsection </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Modern" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modern"> <div class="vector-toc-text"> 11.1 Modern </div> </a> <ul id="toc-Modern-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Historical" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Historical"> <div class="vector-toc-text"> 11.2 Historical </div> </a> <ul id="toc-Historical-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 12 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 13 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Laplace transform</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 61 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-61" aria-hidden="true" > 61 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%8B%E1%8D%95%E1%88%8B%E1%88%B5_%E1%88%BD%E1%8C%8D%E1%8C%8D%E1%88%AD" title="ላፕላስ ሽግግር – Amharic" lang="am" hreflang="am" data-title="ላፕላስ ሽግግር" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%88%D9%8A%D9%84_%D9%84%D8%A7%D8%A8%D9%84%D8%A7%D8%B3" title="تحويل لابلاس – Arabic" lang="ar" hreflang="ar" data-title="تحويل لابلاس" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Tresformada_de_Laplace" title="Tresformada de Laplace – Asturian" lang="ast" hreflang="ast" data-title="Tresformada de Laplace" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B2%E0%A6%BE%E0%A6%AA%E0%A7%8D%E0%A6%B2%E0%A6%BE%E0%A6%B8_%E0%A6%B0%E0%A7%82%E0%A6%AA%E0%A6%BE%E0%A6%A8%E0%A7%8D%E0%A6%A4%E0%A6%B0" title="লাপ্লাস রূপান্তর – Bangla" lang="bn" hreflang="bn" data-title="লাপ্লাস রূপান্তর" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Laplace_pi%C3%A0n-o%C4%81%E2%81%BF" title="Laplace piàn-oāⁿ – Minnan" lang="nan" hreflang="nan" data-title="Laplace piàn-oāⁿ" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5_%D0%BD%D0%B0_%D0%9B%D0%B0%D0%BF%D0%BB%D0%B0%D1%81" title="Преобразование на Лаплас – Bulgarian" lang="bg" hreflang="bg" data-title="Преобразование на Лаплас" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Laplaceova_transformacija" title="Laplaceova transformacija – Bosnian" lang="bs" hreflang="bs" data-title="Laplaceova transformacija" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Transformada_de_Laplace" title="Transformada de Laplace – Catalan" lang="ca" hreflang="ca" data-title="Transformada de Laplace" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Laplaceova_transformace" title="Laplaceova transformace – Czech" lang="cs" hreflang="cs" data-title="Laplaceova transformace" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Laplacetransformation" title="Laplacetransformation – Danish" lang="da" hreflang="da" data-title="Laplacetransformation" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Laplace-Transformation" title="Laplace-Transformation – German" lang="de" hreflang="de" data-title="Laplace-Transformation" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Laplace%27i_teisendus" title="Laplace'i teisendus – Estonian" lang="et" hreflang="et" data-title="Laplace'i teisendus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B5%CF%84%CE%B1%CF%83%CF%87%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CF%83%CE%BC%CF%8C%CF%82_%CE%9B%CE%B1%CF%80%CE%BB%CE%AC%CF%82" title="Μετασχηματισμός Λαπλάς – Greek" lang="el" hreflang="el" data-title="Μετασχηματισμός Λαπλάς" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Transformada_de_Laplace" title="Transformada de Laplace – Spanish" lang="es" hreflang="es" data-title="Transformada de Laplace" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Laplaca_transformo" title="Laplaca transformo – Esperanto" lang="eo" hreflang="eo" data-title="Laplaca transformo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Laplaceren_transformazio" title="Laplaceren transformazio – Basque" lang="eu" hreflang="eu" data-title="Laplaceren transformazio" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A8%D8%AF%DB%8C%D9%84_%D9%84%D8%A7%D9%BE%D9%84%D8%A7%D8%B3" title="تبدیل لاپلاس – Persian" lang="fa" hreflang="fa" data-title="تبدیل لاپلاس" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Transformation_de_Laplace" title="Transformation de Laplace – French" lang="fr" hreflang="fr" data-title="Transformation de Laplace" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Transformada_de_Laplace" title="Transformada de Laplace – Galician" lang="gl" hreflang="gl" data-title="Transformada de Laplace" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/L%C3%A0-ph%C3%BA-l%C3%A0-s%E1%B9%B3%CC%82_pien-von" title="Là-phú-là-sṳ̂ pien-von – Hakka Chinese" lang="hak" hreflang="hak" data-title="Là-phú-là-sṳ̂ pien-von" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="Hakka Chinese" class="interlanguage-link-target">客家語 / Hak-kâ-ngî</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%9D%BC%ED%94%8C%EB%9D%BC%EC%8A%A4_%EB%B3%80%ED%99%98" title="라플라스 변환 – Korean" lang="ko" hreflang="ko" data-title="라플라스 변환" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BC%D5%A1%D5%BA%D5%AC%D5%A1%D5%BD%D5%AB_%D5%B1%D6%87%D5%A1%D6%83%D5%B8%D5%AD%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Լապլասի ձևափոխություն – Armenian" lang="hy" hreflang="hy" data-title="Լապլասի ձևափոխություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B2%E0%A4%BE%E0%A4%AA%E0%A5%8D%E0%A4%B2%E0%A4%BE%E0%A4%B8_%E0%A4%B0%E0%A5%82%E0%A4%AA%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4%E0%A4%B0" title="लाप्लास रूपान्तर – Hindi" lang="hi" hreflang="hi" data-title="लाप्लास रूपान्तर" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Laplaceova_transformacija" title="Laplaceova transformacija – Croatian" lang="hr" hreflang="hr" data-title="Laplaceova transformacija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Transformasi_Laplace" title="Transformasi Laplace – Indonesian" lang="id" hreflang="id" data-title="Transformasi Laplace" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Transformation_de_Laplace" title="Transformation de Laplace – Interlingua" lang="ia" hreflang="ia" data-title="Transformation de Laplace" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Trasformata_di_Laplace" title="Trasformata di Laplace – Italian" lang="it" hreflang="it" data-title="Trasformata di Laplace" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%AA%D7%9E%D7%A8%D7%AA_%D7%9C%D7%A4%D7%9C%D7%A1" title="התמרת לפלס – Hebrew" lang="he" hreflang="he" data-title="התמרת לפלס" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Transformasi_Laplace" title="Transformasi Laplace – Javanese" lang="jv" hreflang="jv" data-title="Transformasi Laplace" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target">Jawa</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9B%D0%B0%D0%BF%D0%BB%D0%B0%D1%81_%D1%82%D2%AF%D1%80%D0%BB%D0%B5%D0%BD%D0%B4%D1%96%D1%80%D1%83%D1%96" title="Лаплас түрлендіруі – Kazakh" lang="kk" hreflang="kk" data-title="Лаплас түрлендіруі" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Laplaso_transformacija" title="Laplaso transformacija – Lithuanian" lang="lt" hreflang="lt" data-title="Laplaso transformacija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Laplace-transzform%C3%A1ci%C3%B3" title="Laplace-transzformáció – Hungarian" lang="hu" hreflang="hu" data-title="Laplace-transzformáció" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B2%E0%B4%BE%E0%B4%AA%E0%B5%8D%E0%B4%B2%E0%B5%87%E0%B4%B8%E0%B5%8D_%E0%B4%AA%E0%B4%B0%E0%B4%BF%E0%B4%B5%E0%B5%BC%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%A8%E0%B4%82" title="ലാപ്ലേസ് പരിവർത്തനം – Malayalam" lang="ml" hreflang="ml" data-title="ലാപ്ലേസ് പരിവർത്തനം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B2%E0%A5%85%E0%A4%AA%E0%A5%8D%E0%A4%B2%E0%A5%87%E0%A4%B8_%E0%A4%AA%E0%A4%B0%E0%A4%BF%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A4%A8" title="लॅप्लेस परिवर्तन – Marathi" lang="mr" hreflang="mr" data-title="लॅप्लेस परिवर्तन" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Laplacetransformatie" title="Laplacetransformatie – Dutch" lang="nl" hreflang="nl" data-title="Laplacetransformatie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A9%E3%83%97%E3%83%A9%E3%82%B9%E5%A4%89%E6%8F%9B" title="ラプラス変換 – Japanese" lang="ja" hreflang="ja" data-title="ラプラス変換" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Laplacetransformasjon" title="Laplacetransformasjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Laplacetransformasjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Laplace-transformasjon" title="Laplace-transformasjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Laplace-transformasjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%94%E1%9F%86%E1%9E%9B%E1%9F%82%E1%9E%84%E1%9E%A1%E1%9E%B6%E1%9E%94%E1%9F%92%E1%9E%9B%E1%9E%B6%E1%9E%9F" title="បំលែងឡាប្លាស – Khmer" lang="km" hreflang="km" data-title="បំលែងឡាប្លាស" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target">ភាសាខ្មែរ</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Trasform%C3%A0_%C3%ABd_Laplace" title="Trasformà ëd Laplace – Piedmontese" lang="pms" hreflang="pms" data-title="Trasformà ëd Laplace" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Transformacja_Laplace%E2%80%99a" title="Transformacja Laplace’a – Polish" lang="pl" hreflang="pl" data-title="Transformacja Laplace’a" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Transformada_de_Laplace" title="Transformada de Laplace – Portuguese" lang="pt" hreflang="pt" data-title="Transformada de Laplace" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Transformat%C4%83_Laplace" title="Transformată Laplace – Romanian" lang="ro" hreflang="ro" data-title="Transformată Laplace" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5_%D0%9B%D0%B0%D0%BF%D0%BB%D0%B0%D1%81%D0%B0" title="Преобразование Лапласа – Russian" lang="ru" hreflang="ru" data-title="Преобразование Лапласа" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Transformimi_i_Laplasit" title="Transformimi i Laplasit – Albanian" lang="sq" hreflang="sq" data-title="Transformimi i Laplasit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Laplace_transform" title="Laplace transform – Simple English" lang="en-simple" hreflang="en-simple" data-title="Laplace transform" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Laplaceova_transformacija" title="Laplaceova transformacija – Slovenian" lang="sl" hreflang="sl" data-title="Laplaceova transformacija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9B%D0%B0%D0%BF%D0%BB%D0%B0%D1%81%D0%BE%D0%B2%D0%B0_%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Лапласова трансформација – Serbian" lang="sr" hreflang="sr" data-title="Лапласова трансформација" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Laplaceova_transformacija" title="Laplaceova transformacija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Laplaceova transformacija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Transformasi_Laplace" title="Transformasi Laplace – Sundanese" lang="su" hreflang="su" data-title="Transformasi Laplace" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Laplace-muunnos" title="Laplace-muunnos – Finnish" lang="fi" hreflang="fi" data-title="Laplace-muunnos" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Laplacetransform" title="Laplacetransform – Swedish" lang="sv" hreflang="sv" data-title="Laplacetransform" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%B2%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%BF%E0%AE%B3%E0%AE%BE%E0%AE%9A%E0%AF%81_%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%81" title="இலப்பிளாசு மாற்று – Tamil" lang="ta" hreflang="ta" data-title="இலப்பிளாசு மாற்று" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tabeddilt_n_Lapla%E1%B9%A3" title="Tabeddilt n Laplaṣ – Kabyle" lang="kab" hreflang="kab" data-title="Tabeddilt n Laplaṣ" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target">Taqbaylit</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%81%E0%B8%9B%E0%B8%A5%E0%B8%87%E0%B8%A5%E0%B8%B2%E0%B8%9B%E0%B8%A5%E0%B8%B1%E0%B8%AA" title="การแปลงลาปลัส – Thai" lang="th" hreflang="th" data-title="การแปลงลาปลัส" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Laplace_d%C3%B6n%C3%BC%C5%9F%C3%BCm%C3%BC" title="Laplace dönüşümü – Turkish" lang="tr" hreflang="tr" data-title="Laplace dönüşümü" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%B5%D1%80%D0%B5%D1%82%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F_%D0%9B%D0%B0%D0%BF%D0%BB%D0%B0%D1%81%D0%B0" title="Перетворення Лапласа – Ukrainian" lang="uk" hreflang="uk" data-title="Перетворення Лапласа" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_bi%E1%BA%BFn_%C4%91%E1%BB%95i_Laplace" title="Phép biến đổi Laplace – Vietnamese" lang="vi" hreflang="vi" data-title="Phép biến đổi Laplace" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E5%8F%98%E6%8D%A2" title="拉普拉斯变换 – Wu" lang="wuu" hreflang="wuu" data-title="拉普拉斯变换" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E8%AE%8A%E6%8F%9B" title="拉普拉斯變換 – Cantonese" lang="yue" hreflang="yue" data-title="拉普拉斯變換" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E5%8F%98%E6%8D%A2" title="拉普拉斯变换 – Chinese" lang="zh" hreflang="zh" data-title="拉普拉斯变换" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q199691#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Integral transform useful in probability theory, physics, and engineering</div> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the Laplace transform, named after <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> (<a href="/wiki/Help:IPA/English" title="Help:IPA/English">/ləˈplɑːs/</a>), is an <a href="/wiki/Integral_transform" title="Integral transform">integral transform</a> that converts a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of a <a href="/wiki/Real_number" title="Real number">real</a> <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a> (usually <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">, in the <a href="/wiki/Time_domain" title="Time domain">time domain</a>) to a function of a <a href="/wiki/Complex_number" title="Complex number">complex</a> variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> (in the complex-valued <a href="/wiki/Frequency_domain" title="Frequency domain">frequency domain</a>, also known as s-domain, or s-plane). The transform is useful for converting <a href="/wiki/Derivative" title="Derivative">differentiation</a> and <a href="/wiki/Integral" title="Integral">integration</a> in the time domain into much easier <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> and <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a> in the Laplace domain (analogous to how <a href="/wiki/Logarithm" title="Logarithm">logarithms</a> are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in <a href="/wiki/Science" title="Science">science</a> and <a href="/wiki/Engineering" title="Engineering">engineering</a>, mostly as a tool for solving linear <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a><a href="#cite_note-Lynn_1986_pp._225–272-1">[1]</a> and <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical systems</a> by simplifying <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a> and <a href="/wiki/Integral_equation" title="Integral equation">integral equations</a> into <a href="/wiki/Algebraic_equation" title="Algebraic equation">algebraic polynomial equations</a>, and by simplifying <a href="/wiki/Convolution" title="Convolution">convolution</a> into <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>.<a href="#cite_note-2">[2]</a><a href="#cite_note-:1-3">[3]</a> Once solved, the inverse Laplace transform reverts to the original domain. The Laplace transform is defined (for suitable functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">) by the <a href="/wiki/Integral" title="Integral">integral</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e651bc0a5afce60c04b04c75ae63d7586da33eb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.908ex; height:5.843ex;" alt="{\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt,}"> where s is a <a href="/wiki/Complex_number" title="Complex number">complex number</a>. It is related to many other transforms, most notably the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> and the <a href="/wiki/Mellin_transform" title="Mellin transform">Mellin transform</a>. <a href="/wiki/Formal_calculation" title="Formal calculation">Formally</a>, the Laplace transform is converted into a Fourier transform by the substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=i\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>i</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=i\omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d22a51fb4400f3d4c9977ab83dce57351807af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.437ex; height:2.176ex;" alt="{\displaystyle s=i\omega }"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"> is real. However, unlike the Fourier transform, which gives the decomposition of a function into its components in each frequency, the Laplace transform of a function with suitable decay is an <a href="/wiki/Analytic_function" title="Analytic function">analytic function</a>, and so has a convergent <a href="/wiki/Power_series" title="Power series">power series</a>, the coefficients of which give the decomposition of a function into its <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moments</a>. Also unlike the Fourier transform, when regarded in this way as an analytic function, the techniques of <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, and especially <a href="/wiki/Contour_integral" class="mw-redirect" title="Contour integral">contour integrals</a>, can be used for calculations. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=1" title="Edit section: History">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Laplace,_Pierre-Simon,_marquis_de.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Laplace%2C_Pierre-Simon%2C_marquis_de.jpg/220px-Laplace%2C_Pierre-Simon%2C_marquis_de.jpg" decoding="async" width="220" height="258" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/3/39/Laplace%2C_Pierre-Simon%2C_marquis_de.jpg 1.5x" data-file-width="256" data-file-height="300" /></a><figcaption>Pierre-Simon, marquis de Laplace</figcaption></figure> The Laplace transform is named after <a href="/wiki/Mathematician" title="Mathematician">mathematician</a> and <a href="/wiki/Astronomer" title="Astronomer">astronomer</a> <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon, Marquis de Laplace</a>, who used a similar transform in his work on <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>.<a href="#cite_note-4">[4]</a> Laplace wrote extensively about the use of <a href="/wiki/Generating_function" title="Generating function">generating functions</a> (1814), and the integral form of the Laplace transform evolved naturally as a result.<a href="#cite_note-5">[5]</a> Laplace's use of generating functions was similar to what is now known as the <a href="/wiki/Z-transform" title="Z-transform">z-transform</a>, and he gave little attention to the <a href="/wiki/Continuous_variable" class="mw-redirect" title="Continuous variable">continuous variable</a> case which was discussed by <a href="/wiki/Niels_Henrik_Abel" title="Niels Henrik Abel">Niels Henrik Abel</a>.<a href="#cite_note-6">[6]</a> From 1744, <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> investigated integrals of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\int X(x)e^{ax}\,dx\quad {\text{ and }}\quad z=\int X(x)x^{A}\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mo>∫</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="1em" /> <mi>z</mi> <mo>=</mo> <mo>∫</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\int X(x)e^{ax}\,dx\quad {\text{ and }}\quad z=\int X(x)x^{A}\,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68755e9371711ae4c72dcba6afb4168c837e675c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.112ex; height:5.676ex;" alt="{\displaystyle z=\int X(x)e^{ax}\,dx\quad {\text{ and }}\quad z=\int X(x)x^{A}\,dx}"> as solutions of differential equations, introducing in particular the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a>.<a href="#cite_note-7">[7]</a> <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a> was an admirer of Euler and, in his work on integrating <a href="/wiki/Probability_density_function" title="Probability density function">probability density functions</a>, investigated expressions of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int X(x)e^{-ax}a^{x}\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>a</mi> <mi>x</mi> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int X(x)e^{-ax}a^{x}\,dx,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e618803bc9a7c33bcbcb907eff4be396836cc51a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.086ex; height:5.676ex;" alt="{\displaystyle \int X(x)e^{-ax}a^{x}\,dx,}"> which resembles a Laplace transform.<a href="#cite_note-8">[8]</a><a href="#cite_note-9">[9]</a> These types of integrals seem first to have attracted Laplace's attention in 1782, where he was following in the spirit of Euler in using the integrals themselves as solutions of equations.<a href="#cite_note-10">[10]</a> However, in 1785, Laplace took the critical step forward when, rather than simply looking for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular. He used an integral of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int x^{s}\varphi (x)\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int x^{s}\varphi (x)\,dx,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2ba1ede95c4ca1e8202c36f5fcf17becc286e8a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.152ex; height:5.676ex;" alt="{\displaystyle \int x^{s}\varphi (x)\,dx,}"> akin to a <a href="/wiki/Mellin_transform" title="Mellin transform">Mellin transform</a>, to transform the whole of a <a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">difference equation</a>, in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.<a href="#cite_note-11">[11]</a> Laplace also recognised that <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a>'s method of <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> for solving the <a href="/wiki/Diffusion_equation" title="Diffusion equation">diffusion equation</a> could only apply to a limited region of space, because those solutions were <a href="/wiki/Periodic_function" title="Periodic function">periodic</a>. In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space.<a href="#cite_note-12">[12]</a> In 1821, <a href="/wiki/Cauchy" class="mw-redirect" title="Cauchy">Cauchy</a> developed an <a href="/wiki/Operational_calculus" title="Operational calculus">operational calculus</a> for the Laplace transform that could be used to study linear differential equations in much the same way the transform is now used in basic engineering. This method was popularized, and perhaps rediscovered, by <a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a> around the turn of the century.<a href="#cite_note-13">[13]</a> <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> used the Laplace transform in his 1859 paper <a href="/wiki/On_the_number_of_primes_less_than_a_given_magnitude" class="mw-redirect" title="On the number of primes less than a given magnitude">On the Number of Primes Less Than a Given Magnitude</a>, in which he also developed the inversion theorem. Riemann used the Laplace transform to develop the functional equation of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>, and this method is still used to related the <a href="/wiki/Modular_form" title="Modular form">modular transformation law</a> of the <a href="/wiki/Jacobi_theta_function" class="mw-redirect" title="Jacobi theta function">Jacobi theta function</a>, which is simple to prove via <a href="/wiki/Poisson_summation" class="mw-redirect" title="Poisson summation">Poisson summation</a>, to the functional equation. <a href="/wiki/Hjalmar_Mellin" title="Hjalmar Mellin">Hjalmar Mellin</a> was among the first to study the Laplace transform, rigorously in the <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a> school of analysis, and apply it to the study of <a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a> and <a href="/wiki/Special_functions" title="Special functions">special functions</a>, at the turn of the 20th century.<a href="#cite_note-14">[14]</a> At around the same time, Heaviside was busy with his operational calculus. <a href="/wiki/Thomas_Joannes_Stieltjes" title="Thomas Joannes Stieltjes">Thomas Joannes Stieltjes</a> considered a generalization of the Laplace transform connected to his <a href="/wiki/Stieltjes_moment_problem" title="Stieltjes moment problem">work on moments</a>. Other contributors in this time period included <a href="/wiki/Mathias_Lerch" title="Mathias Lerch">Mathias Lerch</a>,<a href="#cite_note-15">[15]</a> <a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a>, and <a href="/wiki/Thomas_John_I%27Anson_Bromwich" title="Thomas John I'Anson Bromwich">Thomas Bromwich</a>.<a href="#cite_note-16">[16]</a> In 1934, <a href="/wiki/Raymond_Paley" title="Raymond Paley">Raymond Paley</a> and <a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a> published the important work Fourier transforms in the complex domain, about what is now called the Laplace transform (see below). Also during the 30s, the Laplace transform was instrumental in <a href="/wiki/G_H_Hardy" class="mw-redirect" title="G H Hardy">G H Hardy</a> and <a href="/wiki/John_Edensor_Littlewood" title="John Edensor Littlewood">John Edensor Littlewood</a>'s study of <a href="/wiki/Tauberian_theorem" class="mw-redirect" title="Tauberian theorem">tauberian theorems</a>, and this application was later expounded on by Widder (1941), who developed other aspects of the theory such as a new method for inversion. <a href="/wiki/Edward_Charles_Titchmarsh" title="Edward Charles Titchmarsh">Edward Charles Titchmarsh</a> wrote the influential Introduction to the theory of the Fourier integral (1937). The current widespread use of the transform (mainly in engineering) came about during and soon after <a href="/wiki/World_War_II" title="World War II">World War II</a>,<a href="#cite_note-17">[17]</a> replacing the earlier Heaviside <a href="/wiki/Operational_calculus" title="Operational calculus">operational calculus</a>. The advantages of the Laplace transform had been emphasized by <a href="/wiki/Gustav_Doetsch" title="Gustav Doetsch">Gustav Doetsch</a>,<a href="#cite_note-18">[18]</a> to whom the name Laplace transform is apparently due. <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=2" title="Edit section: Formal definition">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Complex_frequency_s-domain_negative.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Complex_frequency_s-domain_negative.jpg/220px-Complex_frequency_s-domain_negative.jpg" decoding="async" width="220" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Complex_frequency_s-domain_negative.jpg/330px-Complex_frequency_s-domain_negative.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Complex_frequency_s-domain_negative.jpg/440px-Complex_frequency_s-domain_negative.jpg 2x" data-file-width="697" data-file-height="700" /></a><figcaption><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Re (e^{-st})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">ℜ</mi> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Re (e^{-st})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/252b2936f843645deb1da1e8ffbfaf8d581a6c81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.693ex; height:3.009ex;" alt="{\displaystyle \Re (e^{-st})}"> for various complex frequencies in the s-domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s=\sigma +i\omega ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>s</mi> <mo>=</mo> <mi>σ</mi> <mo>+</mo> <mi>i</mi> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (s=\sigma +i\omega ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8fa49e8c0398413251a2f6957f1d57581a22539" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.063ex; height:2.843ex;" alt="{\displaystyle (s=\sigma +i\omega ),}"> which can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\sigma t}\cos(\omega t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>σ</mi> <mi>t</mi> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-\sigma t}\cos(\omega t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ce54deb22d55c09e85f6a4672a99febe8dceac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.368ex; height:3.009ex;" alt="{\displaystyle e^{-\sigma t}\cos(\omega t).}"> The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eb4831f1e0ca1ba7d007dc6b973e54787e1a4b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle \sigma =0}"> axis contains pure cosines. Positive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> contains <a href="/wiki/Damped_sinusoid" class="mw-redirect" title="Damped sinusoid">damped cosines</a>. Negative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> contains <a href="/wiki/Exponential_growth" title="Exponential growth">exponentially growing</a> cosines.</figcaption></figure> The Laplace transform of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> f(t), defined for all <a href="/wiki/Real_number" title="Real number">real numbers</a> t ≥ 0, is the function F(s), which is a unilateral transform defined by <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982f44b7a82326e7cfe6cf9d2aaa945605ae38fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.442ex; height:5.843ex;" alt="{\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt,}">   (Eq. 1) </div> where s is a <a href="/wiki/Complex_number" title="Complex number">complex</a> frequency-domain parameter <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\sigma +i\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>σ</mi> <mo>+</mo> <mi>i</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\sigma +i\omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54cae0a096fc30983395d84e6d07fa3d2818fac0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.607ex; height:2.343ex;" alt="{\displaystyle s=\sigma +i\omega }"> with real numbers σ and ω. An alternate notation for the Laplace transform is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}\{f\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\{f\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2734e92c352d7115a47553097a0e35462f915ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.207ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}\{f\}}"> instead of F,<a href="#cite_note-:1-3">[3]</a> often written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)={\mathcal {L}}\{f(t)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(s)={\mathcal {L}}\{f(t)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd0b873c7673d6ef7f6e9ea4e0e121d5771981f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.595ex; height:2.843ex;" alt="{\displaystyle F(s)={\mathcal {L}}\{f(t)\}}"> in an <a href="/wiki/Function_(mathematics)#Functional_notation" title="Function (mathematics)">abuse of notation</a>. The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that f must be <a href="/wiki/Locally_integrable" class="mw-redirect" title="Locally integrable">locally integrable</a> on [0, ∞). For locally integrable functions that decay at infinity or are of <a href="/wiki/Exponential_type" title="Exponential type">exponential type</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(t)|\leq Ae^{B|t|}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>A</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(t)|\leq Ae^{B|t|}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3afb2261b1eddec57e0578ed13c92cb3a891821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.134ex; height:3.343ex;" alt="{\displaystyle |f(t)|\leq Ae^{B|t|}}">), the integral can be understood to be a (proper) <a href="/wiki/Lebesgue_integral" title="Lebesgue integral">Lebesgue integral</a>. However, for many applications it is necessary to regard it as a <a href="/wiki/Conditionally_convergent" class="mw-redirect" title="Conditionally convergent">conditionally convergent</a> <a href="/wiki/Improper_integral" title="Improper integral">improper integral</a> at ∞. Still more generally, the integral can be understood in a <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">weak sense</a>, and this is dealt with below. One can define the Laplace transform of a finite <a href="/wiki/Borel_measure" title="Borel measure">Borel measure</a> μ by the Lebesgue integral<a href="#cite_note-19">[19]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}\{\mu \}(s)=\int _{[0,\infty )}e^{-st}\,d\mu (t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>μ</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\{\mu \}(s)=\int _{[0,\infty )}e^{-st}\,d\mu (t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d28c2f4979692767dc46b5c537fb4ae7a370118" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.519ex; height:6.009ex;" alt="{\displaystyle {\mathcal {L}}\{\mu \}(s)=\int _{[0,\infty )}e^{-st}\,d\mu (t).}"> An important special case is where μ is a <a href="/wiki/Probability_measure" title="Probability measure">probability measure</a>, for example, the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>. In <a href="/wiki/Operational_calculus" title="Operational calculus">operational calculus</a>, the Laplace transform of a measure is often treated as though the measure came from a probability density function f. In that case, to avoid potential confusion, one often writes <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0^{-}}^{\infty }f(t)e^{-st}\,dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0^{-}}^{\infty }f(t)e^{-st}\,dt,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/927b5a7f96e795abb080680527abae4fc7016cd1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.908ex; height:5.843ex;" alt="{\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0^{-}}^{\infty }f(t)e^{-st}\,dt,}"> where the lower limit of 0− is shorthand notation for <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{\varepsilon \to 0^{+}}\int _{-\varepsilon }^{\infty }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{\varepsilon \to 0^{+}}\int _{-\varepsilon }^{\infty }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49c0048ea0225182048b3f1e705588ef8b4afab2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.807ex; height:6.009ex;" alt="{\displaystyle \lim _{\varepsilon \to 0^{+}}\int _{-\varepsilon }^{\infty }.}"> This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the <a href="/wiki/Laplace%E2%80%93Stieltjes_transform" title="Laplace–Stieltjes transform">Laplace–Stieltjes transform</a>. <div class="mw-heading mw-heading3"><h3 id="Bilateral_Laplace_transform">Bilateral Laplace transform</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=3" title="Edit section: Bilateral Laplace transform">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Two-sided_Laplace_transform" title="Two-sided Laplace transform">Two-sided Laplace transform</a></div> When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is usually intended. The Laplace transform can be alternatively defined as the bilateral Laplace transform, or <a href="/wiki/Two-sided_Laplace_transform" title="Two-sided Laplace transform">two-sided Laplace transform</a>, by extending the limits of integration to be the entire real axis. If that is done, the common unilateral transform simply becomes a special case of the bilateral transform, where the definition of the function being transformed is multiplied by the <a href="/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a>. The bilateral Laplace transform F(s) is defined as follows: <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)=\int _{-\infty }^{\infty }e^{-st}f(t)\,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(s)=\int _{-\infty }^{\infty }e^{-st}f(t)\,dt.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84820999bf8c11f845a98b762047520bf947f960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.549ex; height:6.009ex;" alt="{\displaystyle F(s)=\int _{-\infty }^{\infty }e^{-st}f(t)\,dt.}">   (Eq. 2) </div> An alternate notation for the bilateral Laplace transform is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}\{f\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}\{f\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f162776eba3f1c5666aa134b0dd270b7ce62a998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.147ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}\{f\}}">, instead of F. <div class="mw-heading mw-heading3"><h3 id="Inverse_Laplace_transform">Inverse Laplace transform</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=4" title="Edit section: Inverse Laplace transform">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Inverse_Laplace_transform" title="Inverse Laplace transform">Inverse Laplace transform</a></div> Two integrable functions have the same Laplace transform only if they differ on a set of <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> zero. This means that, on the range of the transform, there is an inverse transform. In fact, besides integrable functions, the Laplace transform is a <a href="/wiki/One-to-one_function" class="mw-redirect" title="One-to-one function">one-to-one mapping</a> from one function space into another in many other function spaces as well, although there is usually no easy characterization of the range. Typical function spaces in which this is true include the spaces of bounded continuous functions, the space <a href="/wiki/Lp_space" title="Lp space">L∞(0, ∞)</a>, or more generally <a href="/wiki/Tempered_distributions" class="mw-redirect" title="Tempered distributions">tempered distributions</a> on (0, ∞). The Laplace transform is also defined and injective for suitable spaces of tempered distributions. In these cases, the image of the Laplace transform lives in a space of <a href="/wiki/Analytic_function" title="Analytic function">analytic functions</a> in the <a href="#Region_of_convergence">region of convergence</a>. The <a href="/wiki/Inverse_Laplace_transform" title="Inverse Laplace transform">inverse Laplace transform</a> is given by the following complex integral, which is known by various names (the Bromwich integral, the Fourier–Mellin integral, and Mellin's inverse formula): <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)={\mathcal {L}}^{-1}\{F\}(t)={\frac {1}{2\pi i}}\lim _{T\to \infty }\int _{\gamma -iT}^{\gamma +iT}e^{st}F(s)\,ds,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">{</mo> <mi>F</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> </mrow> </mfrac> </mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mo>−</mo> <mi>i</mi> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mo>+</mo> <mi>i</mi> <mi>T</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)={\mathcal {L}}^{-1}\{F\}(t)={\frac {1}{2\pi i}}\lim _{T\to \infty }\int _{\gamma -iT}^{\gamma +iT}e^{st}F(s)\,ds,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5efafcd962aa264ad91c3e5dc7b73c07970776a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:47.767ex; height:6.509ex;" alt="{\displaystyle f(t)={\mathcal {L}}^{-1}\{F\}(t)={\frac {1}{2\pi i}}\lim _{T\to \infty }\int _{\gamma -iT}^{\gamma +iT}e^{st}F(s)\,ds,}">   (Eq. 3) </div> where γ is a real number so that the contour path of integration is in the region of convergence of F(s). In most applications, the contour can be closed, allowing the use of the <a href="/wiki/Residue_theorem" title="Residue theorem">residue theorem</a>. An alternative formula for the inverse Laplace transform is given by <a href="/wiki/Post%27s_inversion_formula" class="mw-redirect" title="Post's inversion formula">Post's inversion formula</a>. The limit here is interpreted in the <a href="/wiki/Weak_topology#Weak-*_topology" title="Weak topology">weak-* topology</a>. In practice, it is typically more convenient to decompose a Laplace transform into known transforms of functions obtained from a table and construct the inverse by inspection. <div class="mw-heading mw-heading3"><h3 id="Probability_theory">Probability theory</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=5" title="Edit section: Probability theory">edit</a>]</div> In <a href="/wiki/Probability_theory" title="Probability theory">pure</a> and <a href="/wiki/Applied_probability" title="Applied probability">applied probability</a>, the Laplace transform is defined as an <a href="/wiki/Expected_value" title="Expected value">expected value</a>. If X is a <a href="/wiki/Random_variable" title="Random variable">random variable</a> with probability density function f, then the Laplace transform of f is given by the expectation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}\{f\}(s)=\operatorname {E} \left[e^{-sX}\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>X</mi> </mrow> </msup> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\{f\}(s)=\operatorname {E} \left[e^{-sX}\right],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3da28029aa437333b370a72f00e2745d1f8e8c73" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.14ex; height:3.343ex;" alt="{\displaystyle {\mathcal {L}}\{f\}(s)=\operatorname {E} \left[e^{-sX}\right],}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [r]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>r</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [r]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/715165949e716ef97cb5b61ba0982325cef86a1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.925ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [r]}"> is the <a href="/wiki/Expected_value" title="Expected value">expectation</a> of <a href="/wiki/Random_variable" title="Random variable">random variable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">. By <a href="/wiki/Abuse_of_notation" title="Abuse of notation">convention</a>, this is referred to as the Laplace transform of the random variable X itself. Here, replacing s by −t gives the <a href="/wiki/Moment_generating_function" class="mw-redirect" title="Moment generating function">moment generating function</a> of X. The Laplace transform has applications throughout probability theory, including <a href="/wiki/First_passage_time" class="mw-redirect" title="First passage time">first passage times</a> of <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic processes</a> such as <a href="/wiki/Markov_chain" title="Markov chain">Markov chains</a>, and <a href="/wiki/Renewal_theory" title="Renewal theory">renewal theory</a>. Of particular use is the ability to recover the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of a continuous random variable X by means of the Laplace transform as follows:<a href="#cite_note-20">[20]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}(x)={\mathcal {L}}^{-1}\left\{{\frac {1}{s}}\operatorname {E} \left[e^{-sX}\right]\right\}(x)={\mathcal {L}}^{-1}\left\{{\frac {1}{s}}{\mathcal {L}}\{f\}(s)\right\}(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>X</mi> </mrow> </msup> <mo>]</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(x)={\mathcal {L}}^{-1}\left\{{\frac {1}{s}}\operatorname {E} \left[e^{-sX}\right]\right\}(x)={\mathcal {L}}^{-1}\left\{{\frac {1}{s}}{\mathcal {L}}\{f\}(s)\right\}(x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9728fa6f7b7c0cf6c7fd18917b64a4bbea2c7f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.56ex; height:6.176ex;" alt="{\displaystyle F_{X}(x)={\mathcal {L}}^{-1}\left\{{\frac {1}{s}}\operatorname {E} \left[e^{-sX}\right]\right\}(x)={\mathcal {L}}^{-1}\left\{{\frac {1}{s}}{\mathcal {L}}\{f\}(s)\right\}(x).}"> <div class="mw-heading mw-heading3"><h3 id="Algebraic_construction">Algebraic construction</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=6" title="Edit section: Algebraic construction">edit</a>]</div> The Laplace transform can be alternatively defined in a purely algebraic manner by applying a <a href="/wiki/Field_of_fractions" title="Field of fractions">field of fractions</a> construction to the convolution <a href="/wiki/Ring_(abstract_algebra)" class="mw-redirect" title="Ring (abstract algebra)">ring</a> of functions on the positive half-line. The resulting <a href="/wiki/Convolution_quotient" title="Convolution quotient">space of abstract operators</a> is exactly equivalent to Laplace space, but in this construction the forward and reverse transforms never need to be explicitly defined (avoiding the related difficulties with proving convergence).<a href="#cite_note-21">[21]</a> <div class="mw-heading mw-heading2"><h2 id="Region_of_convergence">Region of convergence</h2>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=7" title="Edit section: Region of convergence">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Pole%E2%80%93zero_plot#Continuous-time_systems" title="Pole–zero plot">Pole–zero plot § Continuous-time systems</a></div> If f is a locally integrable function (or more generally a Borel measure locally of bounded variation), then the Laplace transform F(s) of f converges provided that the limit <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{R\to \infty }\int _{0}^{R}f(t)e^{-st}\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{R\to \infty }\int _{0}^{R}f(t)e^{-st}\,dt}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258b60d39214672b6ca0977ff53c264b1d98d205" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.581ex; height:6.176ex;" alt="{\displaystyle \lim _{R\to \infty }\int _{0}^{R}f(t)e^{-st}\,dt}"> exists. The Laplace transform <a href="/wiki/Absolute_convergence" title="Absolute convergence">converges absolutely</a> if the integral <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }\left|f(t)e^{-st}\right|\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }\left|f(t)e^{-st}\right|\,dt}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29e83ac2f44d3bc78033a4771e78fe1304fed57c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.737ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }\left|f(t)e^{-st}\right|\,dt}"> exists as a proper Lebesgue integral. The Laplace transform is usually understood as <a href="/wiki/Conditional_convergence" title="Conditional convergence">conditionally convergent</a>, meaning that it converges in the former but not in the latter sense. The set of values for which F(s) converges absolutely is either of the form Re(s) > a or Re(s) ≥ a, where a is an <a href="/wiki/Extended_real_number" class="mw-redirect" title="Extended real number">extended real constant</a> with −∞ ≤ a ≤ ∞ (a consequence of the <a href="/wiki/Dominated_convergence_theorem" title="Dominated convergence theorem">dominated convergence theorem</a>). The constant a is known as the abscissa of absolute convergence, and depends on the growth behavior of f(t).<a href="#cite_note-22">[22]</a> Analogously, the two-sided transform converges absolutely in a strip of the form a < Re(s) < b, and possibly including the lines Re(s) = a or Re(s) = b.<a href="#cite_note-23">[23]</a> The subset of values of s for which the Laplace transform converges absolutely is called the region of absolute convergence, or the domain of absolute convergence. In the two-sided case, it is sometimes called the strip of absolute convergence. The Laplace transform is analytic in the region of absolute convergence: this is a consequence of <a href="/wiki/Fubini%27s_theorem" title="Fubini's theorem">Fubini's theorem</a> and <a href="/wiki/Morera%27s_theorem" title="Morera's theorem">Morera's theorem</a>. Similarly, the set of values for which F(s) converges (conditionally or absolutely) is known as the region of conditional convergence, or simply the region of convergence (ROC). If the Laplace transform converges (conditionally) at s = s0, then it automatically converges for all s with Re(s) > Re(s0). Therefore, the region of convergence is a half-plane of the form Re(s) > a, possibly including some points of the boundary line Re(s) = a. In the region of convergence Re(s) > Re(s0), the Laplace transform of f can be expressed by <a href="/wiki/Integration_by_parts" title="Integration by parts">integrating by parts</a> as the integral <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)=(s-s_{0})\int _{0}^{\infty }e^{-(s-s_{0})t}\beta (t)\,dt,\quad \beta (u)=\int _{0}^{u}e^{-s_{0}t}f(t)\,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>t</mi> </mrow> </msup> <mi>β</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> <mspace width="1em" /> <mi>β</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(s)=(s-s_{0})\int _{0}^{\infty }e^{-(s-s_{0})t}\beta (t)\,dt,\quad \beta (u)=\int _{0}^{u}e^{-s_{0}t}f(t)\,dt.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99e8feba04b19cd2c61d80c1ed00bed66332bc6e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:62.038ex; height:5.843ex;" alt="{\displaystyle F(s)=(s-s_{0})\int _{0}^{\infty }e^{-(s-s_{0})t}\beta (t)\,dt,\quad \beta (u)=\int _{0}^{u}e^{-s_{0}t}f(t)\,dt.}"> That is, F(s) can effectively be expressed, in the region of convergence, as the absolutely convergent Laplace transform of some other function. In particular, it is analytic. There are several <a href="/wiki/Paley%E2%80%93Wiener_theorem" title="Paley–Wiener theorem">Paley–Wiener theorems</a> concerning the relationship between the decay properties of f, and the properties of the Laplace transform within the region of convergence. In engineering applications, a function corresponding to a <a href="/wiki/Linear_time-invariant_system" title="Linear time-invariant system">linear time-invariant (LTI) system</a> is stable if every bounded input produces a bounded output. This is equivalent to the absolute convergence of the Laplace transform of the impulse response function in the region Re(s) ≥ 0. As a result, LTI systems are stable, provided that the poles of the Laplace transform of the impulse response function have negative real part. This ROC is used in knowing about the causality and stability of a system. <div class="mw-heading mw-heading2"><h2 id="Properties_and_theorems">Properties and theorems</h2>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=8" title="Edit section: Properties and theorems">edit</a>]</div> The Laplace transform's key property is that it converts <a href="/wiki/Derivative" title="Derivative">differentiation</a> and <a href="/wiki/Integral" title="Integral">integration</a> in the time domain into multiplication and division by s in the Laplace domain. Thus, the Laplace variable s is also known as an operator variable in the Laplace domain: either the derivative operator or (for s−1) the integration operator. Given the functions f(t) and g(t), and their respective Laplace transforms F(s) and G(s), <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f(t)&={\mathcal {L}}^{-1}\{F(s)\},\\g(t)&={\mathcal {L}}^{-1}\{G(s)\},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">{</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">{</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(t)&={\mathcal {L}}^{-1}\{F(s)\},\\g(t)&={\mathcal {L}}^{-1}\{G(s)\},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f0bc42b6abe9cf8011aaa7c74bb3d4f64006b3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.412ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}f(t)&={\mathcal {L}}^{-1}\{F(s)\},\\g(t)&={\mathcal {L}}^{-1}\{G(s)\},\end{aligned}}}"> the following table is a list of properties of unilateral Laplace transform:<a href="#cite_note-24">[24]</a> <table class="wikitable" id="291017_tableid"> <caption>Properties of the unilateral Laplace transform </caption> <tbody><tr> <th scope="col">Property </th> <th scope="col">Time domain </th> <th scope="col">s domain </th> <th scope="col">Comment </th></tr> <tr> <th scope="row"><a href="/wiki/Linearity" title="Linearity">Linearity</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle af(t)+bg(t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle af(t)+bg(t)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c439e5ee4a098d215f3da4140f07fe1cc1c8a38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.341ex; height:2.843ex;" alt="{\displaystyle af(t)+bg(t)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aF(s)+bG(s)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mi>G</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aF(s)+bG(s)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/088f5665dc6da7d301477215f45387b8d60d1567" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.015ex; height:2.843ex;" alt="{\displaystyle aF(s)+bG(s)\ }"> </td> <td>Can be proved using basic rules of integration. </td></tr> <tr> <th scope="row">Frequency-domain derivative </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle tf(t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle tf(t)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c4cc37f83e8dd4af082f40b6e4d8f6c9cae257d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.348ex; height:2.843ex;" alt="{\displaystyle tf(t)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -F'(s)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -F'(s)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5daabdcf623d228093dd62c7fa3eebdb4171e360" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.788ex; height:3.009ex;" alt="{\displaystyle -F'(s)\ }"> </td> <td>F′ is the first derivative of F with respect to s. </td></tr> <tr> <th scope="row">Frequency-domain general derivative </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t^{n}f(t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t^{n}f(t)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10fbe602f1730ce9da2c17500fefec4f4b2f3fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.566ex; height:2.843ex;" alt="{\displaystyle t^{n}f(t)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{n}F^{(n)}(s)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)^{n}F^{(n)}(s)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8286f5111f20c15a32de3f69af0f0aee5e90effa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.791ex; height:3.343ex;" alt="{\displaystyle (-1)^{n}F^{(n)}(s)\ }"> </td> <td>More general form, nth derivative of F(s). </td></tr> <tr> <th scope="row"><a href="/wiki/Derivative" title="Derivative">Derivative</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(t)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9694560ffc3e47b35c1c3e1412477c27b320b9fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.235ex; height:3.009ex;" alt="{\displaystyle f'(t)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle sF(s)-f(0^{-})\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle sF(s)-f(0^{-})\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5885b8f8a869ee62abce04481b8e8459c8ac35a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.913ex; height:3.009ex;" alt="{\displaystyle sF(s)-f(0^{-})\ }"> </td> <td>f is assumed to be a <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable function</a>, and its derivative is assumed to be of exponential type. This can then be obtained by integration by parts </td></tr> <tr> <th scope="row">Second derivative </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f''(t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f''(t)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1eff2b5586780c5c53bea1ee26d4af2b15c3ee0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.687ex; height:3.009ex;" alt="{\displaystyle f''(t)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle s^{2}F(s)-sf(0^{-})-f'(0^{-})\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>s</mi> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle s^{2}F(s)-sf(0^{-})-f'(0^{-})\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37b9eaf40cb443a2c296c603317a7ce71fe9feb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.386ex; height:3.009ex;" alt="{\textstyle s^{2}F(s)-sf(0^{-})-f'(0^{-})\ }"> </td> <td>f is assumed twice differentiable and the second derivative to be of exponential type. Follows by applying the Differentiation property to f′(t). </td></tr> <tr> <th scope="row">General derivative </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{(n)}(t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{(n)}(t)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a044ef6802b999bd3e6361c2c40d48d41b29f714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.048ex; height:3.343ex;" alt="{\displaystyle f^{(n)}(t)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{n}F(s)-\sum _{k=1}^{n}s^{n-k}f^{(k-1)}(0^{-})\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s^{n}F(s)-\sum _{k=1}^{n}s^{n-k}f^{(k-1)}(0^{-})\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a0f4ab85560ae8f128c86c5dec506d9a15fb97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.828ex; height:6.843ex;" alt="{\displaystyle s^{n}F(s)-\sum _{k=1}^{n}s^{n-k}f^{(k-1)}(0^{-})\ }"> </td> <td>f is assumed to be n-times differentiable, with nth derivative of exponential type. Follows by <a href="/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a>. </td></tr> <tr> <th scope="row">Frequency-domain <a href="/wiki/Integral" title="Integral">integration</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{t}}f(t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>t</mi> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{t}}f(t)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df18dc76a9f95733731da12ab2d34a113a094e1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.507ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{t}}f(t)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{s}^{\infty }F(\sigma )\,d\sigma \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>F</mi> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>σ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{s}^{\infty }F(\sigma )\,d\sigma \ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40003b07a307c6bb60c8f43d4d9a92ea983c8d0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.12ex; height:5.843ex;" alt="{\displaystyle \int _{s}^{\infty }F(\sigma )\,d\sigma \ }"> </td> <td>This is deduced using the nature of frequency differentiation and conditional convergence. </td></tr> <tr> <th scope="row">Time-domain integration </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{t}f(\tau )\,d\tau =(u*f)(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>∗</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{t}f(\tau )\,d\tau =(u*f)(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43c4fa94ab0e694f4085849ae2c36110bd7d71ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.132ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{t}f(\tau )\,d\tau =(u*f)(t)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1 \over s}F(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {1 \over s}F(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/223581ba83d4b37e90d73631a211033a49cce039" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.639ex; height:5.176ex;" alt="{\displaystyle {1 \over s}F(s)}"> </td> <td>u(t) is the Heaviside step function and (u ∗ f)(t) is the <a href="/wiki/Convolution" title="Convolution">convolution</a> of u(t) and f(t). </td></tr> <tr> <th scope="row">Frequency shifting </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{at}f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>t</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{at}f(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44bf3375acb0a610c819b37bdd76a6be8390efbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.707ex; height:3.009ex;" alt="{\displaystyle e^{at}f(t)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s-a)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(s-a)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30de22ed8ce5737d22599626e896e0d062cf0cd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.291ex; height:2.843ex;" alt="{\displaystyle F(s-a)\ }"> </td> <td> </td></tr> <tr> <th scope="row">Time shifting </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t-a)u(t-a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t-a)u(t-a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13519ec40556e7d87ce3b7cd60ea8363a4cb4068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.047ex; height:2.843ex;" alt="{\displaystyle f(t-a)u(t-a)}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)u(t-a)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)u(t-a)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/428db0e16e0decf4c15d7e40cfd011bc76eb8013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.557ex; height:2.843ex;" alt="{\displaystyle f(t)u(t-a)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-as}F(s)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>a</mi> <mi>s</mi> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-as}F(s)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e79cbddac7b358b46a80bce893de401ac67b604" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.456ex; height:3.009ex;" alt="{\displaystyle e^{-as}F(s)\ }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-as}{\mathcal {L}}\{f(t+a)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>a</mi> <mi>s</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-as}{\mathcal {L}}\{f(t+a)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1422f7273ecc0af7ab0332086db1394d1a4e1553" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.161ex; height:3.009ex;" alt="{\displaystyle e^{-as}{\mathcal {L}}\{f(t+a)\}}"> </td> <td>a > 0, u(t) is the Heaviside step function </td></tr> <tr> <th scope="row">Time scaling </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(at)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(at)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc5e841ccc8ab8cda90ccd73dcd798c3d9cca921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.157ex; height:2.843ex;" alt="{\displaystyle f(at)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{a}}F\left({s \over a}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> <mi>F</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{a}}F\left({s \over a}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ed97406d1e31926551099511e21be324562e7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.035ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{a}}F\left({s \over a}\right)}"> </td> <td>a > 0 </td></tr> <tr> <th scope="row"><a href="/wiki/Multiplication" title="Multiplication">Multiplication</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)g(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)g(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1909fe4283b4370ed5330e44605ae1b3ef7ce411" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.692ex; height:2.843ex;" alt="{\displaystyle f(t)g(t)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2\pi i}}\lim _{T\to \infty }\int _{c-iT}^{c+iT}F(\sigma )G(s-\sigma )\,d\sigma \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> </mrow> </mfrac> </mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>−</mo> <mi>i</mi> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>+</mo> <mi>i</mi> <mi>T</mi> </mrow> </msubsup> <mi>F</mi> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>σ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2\pi i}}\lim _{T\to \infty }\int _{c-iT}^{c+iT}F(\sigma )G(s-\sigma )\,d\sigma \ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c430d083a7fe87a871ed5a0228c8adf199be4316" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.439ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{2\pi i}}\lim _{T\to \infty }\int _{c-iT}^{c+iT}F(\sigma )G(s-\sigma )\,d\sigma \ }"> </td> <td>The integration is done along the vertical line Re(σ) = c that lies entirely within the region of convergence of F.<a href="#cite_note-25">[25]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Convolution" title="Convolution">Convolution</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d53844ff3560228f16c23b6c67488724fb09330" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.725ex; height:6.176ex;" alt="{\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)\cdot G(s)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(s)\cdot G(s)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/200e8a1e18b17f2311b7a19d6f1eac9949983c84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.627ex; height:2.843ex;" alt="{\displaystyle F(s)\cdot G(s)\ }"> </td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Circular_convolution" title="Circular convolution">Circular convolution</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(t)=\int _{0}^{T}f(\tau )g(t-\tau )\,d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g)(t)=\int _{0}^{T}f(\tau )g(t-\tau )\,d\tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40246c062f029ca53b2da533dbfb3497aa3911c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.288ex; height:6.176ex;" alt="{\displaystyle (f*g)(t)=\int _{0}^{T}f(\tau )g(t-\tau )\,d\tau }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)\cdot G(s)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(s)\cdot G(s)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/200e8a1e18b17f2311b7a19d6f1eac9949983c84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.627ex; height:2.843ex;" alt="{\displaystyle F(s)\cdot G(s)\ }"> </td> <td>For periodic functions with period T. </td></tr> <tr> <th scope="row"><a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">Complex conjugation</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ffa26d064d965ec25dea445401e03fe9044a40f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.024ex; height:2.843ex;" alt="{\displaystyle f^{*}(t)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{*}(s^{*})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F^{*}(s^{*})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a4b453480cfe4e42440787d372726edde21eef1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.823ex; height:2.843ex;" alt="{\displaystyle F^{*}(s^{*})}"> </td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Periodic_function" title="Periodic function">Periodic function</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1 \over 1-e^{-Ts}}\int _{0}^{T}e^{-st}f(t)\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>T</mi> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {1 \over 1-e^{-Ts}}\int _{0}^{T}e^{-st}f(t)\,dt}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8ff2c2aeeae11c1ec1349d21d45e054ab59369" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.318ex; height:6.176ex;" alt="{\displaystyle {1 \over 1-e^{-Ts}}\int _{0}^{T}e^{-st}f(t)\,dt}"> </td> <td>f(t) is a periodic function of period T so that f(t) = f(t + T), for all t ≥ 0. This is the result of the time shifting property and the <a href="/wiki/Geometric_series" title="Geometric series">geometric series</a>. </td></tr> <tr> <th scope="row"><a href="/wiki/Periodic_summation" title="Periodic summation">Periodic summation</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{P}(t)=\sum _{n=0}^{\infty }f(t-Tn)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>T</mi> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{P}(t)=\sum _{n=0}^{\infty }f(t-Tn)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/564e45d80f9562947ca13f953fd511c0989b26e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.894ex; height:6.843ex;" alt="{\displaystyle f_{P}(t)=\sum _{n=0}^{\infty }f(t-Tn)}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{P}(t)=\sum _{n=0}^{\infty }(-1)^{n}f(t-Tn)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>T</mi> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{P}(t)=\sum _{n=0}^{\infty }(-1)^{n}f(t-Tn)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daf13571be8be59de5dd706c12e67953794fa9cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.505ex; height:6.843ex;" alt="{\displaystyle f_{P}(t)=\sum _{n=0}^{\infty }(-1)^{n}f(t-Tn)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{P}(s)={\frac {1}{1-e^{-Ts}}}F(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>T</mi> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{P}(s)={\frac {1}{1-e^{-Ts}}}F(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73b862401eb0c4d4a8b798f11dc0151f72e07ddc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.961ex; height:5.676ex;" alt="{\displaystyle F_{P}(s)={\frac {1}{1-e^{-Ts}}}F(s)}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{P}(s)={\frac {1}{1+e^{-Ts}}}F(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>T</mi> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{P}(s)={\frac {1}{1+e^{-Ts}}}F(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e26c58ab0fd3d2b139bb419b175bb35e606e650a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.961ex; height:5.676ex;" alt="{\displaystyle F_{P}(s)={\frac {1}{1+e^{-Ts}}}F(s)}"> </td> <td> </td></tr></tbody></table> <dl><dt><a href="/wiki/Initial_value_theorem" title="Initial value theorem">Initial value theorem</a></dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0^{+})=\lim _{s\to \infty }{sF(s)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(0^{+})=\lim _{s\to \infty }{sF(s)}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74ac05ae82e8d1284a536cb8950938af4b8faecc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.682ex; height:4.009ex;" alt="{\displaystyle f(0^{+})=\lim _{s\to \infty }{sF(s)}.}"></dd> <dt><a href="/wiki/Final_value_theorem" title="Final value theorem">Final value theorem</a></dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\infty )=\lim _{s\to 0}{sF(s)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\infty )=\lim _{s\to 0}{sF(s)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6585a27c398148bf0dd5f3d6bdbeb7264af9a216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.864ex; height:4.009ex;" alt="{\displaystyle f(\infty )=\lim _{s\to 0}{sF(s)}}">, if all <a href="/wiki/Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">poles</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle sF(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle sF(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc7c6e65d7dafda3220815261abddc3ea392b03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.731ex; height:2.843ex;" alt="{\displaystyle sF(s)}"> are in the left half-plane.</dd> <dd>The final value theorem is useful because it gives the long-term behaviour without having to perform <a href="/wiki/Partial_fraction" class="mw-redirect" title="Partial fraction">partial fraction</a> decompositions (or other difficult algebra). If F(s) has a pole in the right-hand plane or poles on the imaginary axis (e.g., if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=e^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=e^{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15fa4a9ed9d37ae49a8debc38fb8c82b737c25b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.935ex; height:3.009ex;" alt="{\displaystyle f(t)=e^{t}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=\sin(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=\sin(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b93d4577f832ee44bf5fe84880495a114bfe3c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.53ex; height:2.843ex;" alt="{\displaystyle f(t)=\sin(t)}">), then the behaviour of this formula is undefined.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Relation_to_power_series">Relation to power series</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=9" title="Edit section: Relation to power series">edit</a>]</div> The Laplace transform can be viewed as a <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> analogue of a <a href="/wiki/Power_series" title="Power series">power series</a>.<a href="#cite_note-26">[26]</a> If a(n) is a discrete function of a positive integer n, then the power series associated to a(n) is the series <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }a(n)x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>a</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }a(n)x^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df542255cd9b38f9a353c2f2be5fb3a90aaa3002" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.724ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }a(n)x^{n}}"> where x is a real variable (see <a href="/wiki/Z-transform" title="Z-transform">Z-transform</a>). Replacing summation over n with integration over t, a continuous version of the power series becomes <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }f(t)x^{t}\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }f(t)x^{t}\,dt}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cac2b54c4539736f9b6d0142a168b97b25ecfd4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.252ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }f(t)x^{t}\,dt}"> where the discrete function a(n) is replaced by the continuous one f(t). Changing the base of the power from x to e gives <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }f(t)\left(e^{\ln {x}}\right)^{t}\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }f(t)\left(e^{\ln {x}}\right)^{t}\,dt}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5eaf1980963132ec4399741805fb6b8126143b2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.067ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }f(t)\left(e^{\ln {x}}\right)^{t}\,dt}"> For this to converge for, say, all bounded functions f, it is necessary to require that ln x < 0. Making the substitution −s = ln x gives just the Laplace transform: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }f(t)e^{-st}\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }f(t)e^{-st}\,dt}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f1ac056ca6fa1e2a6eceaa4cc66eba57effc7dc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.056ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }f(t)e^{-st}\,dt}"> In other words, the Laplace transform is a continuous analog of a power series, in which the discrete parameter n is replaced by the continuous parameter t, and x is replaced by e−s. <div class="mw-heading mw-heading3"><h3 id="Relation_to_moments">Relation to moments</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=10" title="Edit section: Relation to moments">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Moment-generating_function" title="Moment-generating function">Moment-generating function</a></div> The quantities <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{n}=\int _{0}^{\infty }t^{n}f(t)\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{n}=\int _{0}^{\infty }t^{n}f(t)\,dt}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1f33c010791e74ba2ca9856005ad67cccd27dd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.873ex; height:5.843ex;" alt="{\displaystyle \mu _{n}=\int _{0}^{\infty }t^{n}f(t)\,dt}"> are the moments of the function f. If the first n moments of f converge absolutely, then by repeated <a href="/wiki/Differentiation_under_the_integral" class="mw-redirect" title="Differentiation under the integral">differentiation under the integral</a>, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{n}({\mathcal {L}}f)^{(n)}(0)=\mu _{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mi>f</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)^{n}({\mathcal {L}}f)^{(n)}(0)=\mu _{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bacd14dcf2df4497c46ddae5b051cbe2f3f869e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.525ex; height:3.343ex;" alt="{\displaystyle (-1)^{n}({\mathcal {L}}f)^{(n)}(0)=\mu _{n}.}"> This is of special significance in probability theory, where the moments of a random variable X are given by the expectation values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{n}=\operatorname {E} [X^{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{n}=\operatorname {E} [X^{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a470b23edb981c95e00f5690370662b6b102341" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.81ex; height:2.843ex;" alt="{\displaystyle \mu _{n}=\operatorname {E} [X^{n}]}">. Then, the relation holds <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{n}=(-1)^{n}{\frac {d^{n}}{ds^{n}}}\operatorname {E} \left[e^{-sX}\right](0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>X</mi> </mrow> </msup> <mo>]</mo> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{n}=(-1)^{n}{\frac {d^{n}}{ds^{n}}}\operatorname {E} \left[e^{-sX}\right](0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aebff76a8e2a99d41a4d654999b7309d1a2f5916" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.371ex; height:5.509ex;" alt="{\displaystyle \mu _{n}=(-1)^{n}{\frac {d^{n}}{ds^{n}}}\operatorname {E} \left[e^{-sX}\right](0).}"> <div class="mw-heading mw-heading3"><h3 id="Transform_of_a_function's_derivative">Transform of a function's derivative</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=11" title="Edit section: Transform of a function's derivative">edit</a>]</div> It is often convenient to use the differentiation property of the Laplace transform to find the transform of a function's derivative. This can be derived from the basic expression for a Laplace transform as follows: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\mathcal {L}}\left\{f(t)\right\}&=\int _{0^{-}}^{\infty }e^{-st}f(t)\,dt\\[6pt]&=\left[{\frac {f(t)e^{-st}}{-s}}\right]_{0^{-}}^{\infty }-\int _{0^{-}}^{\infty }{\frac {e^{-st}}{-s}}f'(t)\,dt\quad {\text{(by parts)}}\\[6pt]&=\left[-{\frac {f(0^{-})}{-s}}\right]+{\frac {1}{s}}{\mathcal {L}}\left\{f'(t)\right\},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> </mrow> <mrow> <mo>−</mo> <mi>s</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mrow> <mo>−</mo> <mi>s</mi> </mrow> </mfrac> </mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>(by parts)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>−</mo> <mi>s</mi> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {L}}\left\{f(t)\right\}&=\int _{0^{-}}^{\infty }e^{-st}f(t)\,dt\\[6pt]&=\left[{\frac {f(t)e^{-st}}{-s}}\right]_{0^{-}}^{\infty }-\int _{0^{-}}^{\infty }{\frac {e^{-st}}{-s}}f'(t)\,dt\quad {\text{(by parts)}}\\[6pt]&=\left[-{\frac {f(0^{-})}{-s}}\right]+{\frac {1}{s}}{\mathcal {L}}\left\{f'(t)\right\},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/574460c0fe29a550c4b4a66b9aa82601ea9f956b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.505ex; width:57.406ex; height:22.176ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {L}}\left\{f(t)\right\}&=\int _{0^{-}}^{\infty }e^{-st}f(t)\,dt\\[6pt]&=\left[{\frac {f(t)e^{-st}}{-s}}\right]_{0^{-}}^{\infty }-\int _{0^{-}}^{\infty }{\frac {e^{-st}}{-s}}f'(t)\,dt\quad {\text{(by parts)}}\\[6pt]&=\left[-{\frac {f(0^{-})}{-s}}\right]+{\frac {1}{s}}{\mathcal {L}}\left\{f'(t)\right\},\end{aligned}}}"> yielding <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}\{f'(t)\}=s\cdot {\mathcal {L}}\{f(t)\}-f(0^{-}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>s</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\{f'(t)\}=s\cdot {\mathcal {L}}\{f(t)\}-f(0^{-}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98dfcd64d59743af64df17e66bcc60208ded9726" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.555ex; height:3.009ex;" alt="{\displaystyle {\mathcal {L}}\{f'(t)\}=s\cdot {\mathcal {L}}\{f(t)\}-f(0^{-}),}"> and in the bilateral case, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}\{f'(t)\}=s\int _{-\infty }^{\infty }e^{-st}f(t)\,dt=s\cdot {\mathcal {L}}\{f(t)\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>s</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>s</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\{f'(t)\}=s\int _{-\infty }^{\infty }e^{-st}f(t)\,dt=s\cdot {\mathcal {L}}\{f(t)\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6180c907cad4b6aeb6d20aee7df0b3b13c32511" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.693ex; height:6.009ex;" alt="{\displaystyle {\mathcal {L}}\{f'(t)\}=s\int _{-\infty }^{\infty }e^{-st}f(t)\,dt=s\cdot {\mathcal {L}}\{f(t)\}.}"> The general result <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}\left\{f^{(n)}(t)\right\}=s^{n}\cdot {\mathcal {L}}\{f(t)\}-s^{n-1}f(0^{-})-\cdots -f^{(n-1)}(0^{-}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mo>⋯</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\left\{f^{(n)}(t)\right\}=s^{n}\cdot {\mathcal {L}}\{f(t)\}-s^{n-1}f(0^{-})-\cdots -f^{(n-1)}(0^{-}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577cda357519bb4a7797adcac4ae094235d03d7c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.965ex; height:4.843ex;" alt="{\displaystyle {\mathcal {L}}\left\{f^{(n)}(t)\right\}=s^{n}\cdot {\mathcal {L}}\{f(t)\}-s^{n-1}f(0^{-})-\cdots -f^{(n-1)}(0^{-}),}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{(n)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{(n)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dfb1963ccde0e87eb3838f51dc19041e2ff3816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.818ex; height:3.176ex;" alt="{\displaystyle f^{(n)}}"> denotes the nth derivative of f, can then be established with an inductive argument. <div class="mw-heading mw-heading3"><h3 id="Evaluating_integrals_over_the_positive_real_axis">Evaluating integrals over the positive real axis</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=12" title="Edit section: Evaluating integrals over the positive real axis">edit</a>]</div> A useful property of the Laplace transform is the following: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }f(x)g(x)\,dx=\int _{0}^{\infty }({\mathcal {L}}f)(s)\cdot ({\mathcal {L}}^{-1}g)(s)\,ds}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }f(x)g(x)\,dx=\int _{0}^{\infty }({\mathcal {L}}f)(s)\cdot ({\mathcal {L}}^{-1}g)(s)\,ds}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f622261e2084edd910d0fa39e34e454165c4734" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.495ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }f(x)g(x)\,dx=\int _{0}^{\infty }({\mathcal {L}}f)(s)\cdot ({\mathcal {L}}^{-1}g)(s)\,ds}"> under suitable assumptions on the behaviour of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b6ab1762925585cd7605809caa8b1b5284177b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.429ex; height:2.509ex;" alt="{\displaystyle f,g}"> in a right neighbourhood of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> and on the decay rate of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b6ab1762925585cd7605809caa8b1b5284177b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.429ex; height:2.509ex;" alt="{\displaystyle f,g}"> in a left neighbourhood of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }">. The above formula is a variation of integration by parts, with the operators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e80537df4f7e8d3d157ed7d50514cfe0c04b91f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.382ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a665e507c57994476eade35eace082d3859f36b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:5.513ex; height:5.676ex;" alt="{\displaystyle \int \,dx}"> being replaced by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a0067e9f5f25b8a53746c33679f86814cfb61b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.937ex; height:2.676ex;" alt="{\displaystyle {\mathcal {L}}^{-1}}">. Let us prove the equivalent formulation: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }({\mathcal {L}}f)(x)g(x)\,dx=\int _{0}^{\infty }f(s)({\mathcal {L}}g)(s)\,ds.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }({\mathcal {L}}f)(x)g(x)\,dx=\int _{0}^{\infty }f(s)({\mathcal {L}}g)(s)\,ds.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94e11a828b41fc154473863652db18508b5b9881" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.13ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }({\mathcal {L}}f)(x)g(x)\,dx=\int _{0}^{\infty }f(s)({\mathcal {L}}g)(s)\,ds.}"> By plugging in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {L}}f)(x)=\int _{0}^{\infty }f(s)e^{-sx}\,ds}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathcal {L}}f)(x)=\int _{0}^{\infty }f(s)e^{-sx}\,ds}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/508ae153cca088e9e9de4a25d6d250ff555fd9a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.833ex; height:5.843ex;" alt="{\displaystyle ({\mathcal {L}}f)(x)=\int _{0}^{\infty }f(s)e^{-sx}\,ds}"> the left-hand side turns into: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }\int _{0}^{\infty }f(s)g(x)e^{-sx}\,ds\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }\int _{0}^{\infty }f(s)g(x)e^{-sx}\,ds\,dx,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6086e6b1dd955c83c09298d6ed8a3431686848c7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.465ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }\int _{0}^{\infty }f(s)g(x)e^{-sx}\,ds\,dx,}"> but assuming Fubini's theorem holds, by reversing the order of integration we get the wanted right-hand side. This method can be used to compute integrals that would otherwise be difficult to compute using elementary methods of real calculus. For example, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }{\frac {\sin x}{x}}dx=\int _{0}^{\infty }{\mathcal {L}}(1)(x)\sin xdx=\int _{0}^{\infty }1\cdot {\mathcal {L}}(\sin )(x)dx=\int _{0}^{\infty }{\frac {dx}{x^{2}+1}}={\frac {\pi }{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mi>x</mi> </mfrac> </mrow> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mn>1</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>sin</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }{\frac {\sin x}{x}}dx=\int _{0}^{\infty }{\mathcal {L}}(1)(x)\sin xdx=\int _{0}^{\infty }1\cdot {\mathcal {L}}(\sin )(x)dx=\int _{0}^{\infty }{\frac {dx}{x^{2}+1}}={\frac {\pi }{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4ccab9798f6d3f06ff673e67b18de97e184465" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:79.307ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }{\frac {\sin x}{x}}dx=\int _{0}^{\infty }{\mathcal {L}}(1)(x)\sin xdx=\int _{0}^{\infty }1\cdot {\mathcal {L}}(\sin )(x)dx=\int _{0}^{\infty }{\frac {dx}{x^{2}+1}}={\frac {\pi }{2}}.}"> <div class="mw-heading mw-heading2"><h2 id="Relationship_to_other_transforms">Relationship to other transforms</h2>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=13" title="Edit section: Relationship to other transforms">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Laplace–Stieltjes_transform">Laplace–Stieltjes transform</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=14" title="Edit section: Laplace–Stieltjes transform">edit</a>]</div> The (unilateral) Laplace–Stieltjes transform of a function g : ℝ → ℝ is defined by the <a href="/wiki/Lebesgue%E2%80%93Stieltjes_integral" class="mw-redirect" title="Lebesgue–Stieltjes integral">Lebesgue–Stieltjes integral</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\mathcal {L}}^{*}g\}(s)=\int _{0}^{\infty }e^{-st}\,d\,g(t)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>g</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mspace width="thinmathspace" /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\mathcal {L}}^{*}g\}(s)=\int _{0}^{\infty }e^{-st}\,d\,g(t)~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3158842e61c2dda2beea2744acf283989e59d2a4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.765ex; height:5.843ex;" alt="{\displaystyle \{{\mathcal {L}}^{*}g\}(s)=\int _{0}^{\infty }e^{-st}\,d\,g(t)~.}"> The function g is assumed to be of <a href="/wiki/Bounded_variation" title="Bounded variation">bounded variation</a>. If g is the <a href="/wiki/Antiderivative" title="Antiderivative">antiderivative</a> of f: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=\int _{0}^{x}f(t)\,d\,t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mspace width="thinmathspace" /> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=\int _{0}^{x}f(t)\,d\,t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e19fc1607e09b034f776496c7a1a0398424b70c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.134ex; height:5.843ex;" alt="{\displaystyle g(x)=\int _{0}^{x}f(t)\,d\,t}"> then the Laplace–Stieltjes transform of g and the Laplace transform of f coincide. In general, the Laplace–Stieltjes transform is the Laplace transform of the <a href="/wiki/Stieltjes_measure" class="mw-redirect" title="Stieltjes measure">Stieltjes measure</a> associated to g. So in practice, the only distinction between the two transforms is that the Laplace transform is thought of as operating on the density function of the measure, whereas the Laplace–Stieltjes transform is thought of as operating on its <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a>.<a href="#cite_note-27">[27]</a> <div class="mw-heading mw-heading3"><h3 id="Fourier_transform">Fourier transform</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=15" title="Edit section: Fourier transform">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Fourier_transform#Laplace_transform" title="Fourier transform">Fourier transform § Laplace transform</a></div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> be a complex-valued Lebesgue integrable function supported on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\infty )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc2d914c2df66bc0f7893bfb8da36766650fe47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle [0,\infty )}">, and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)={\mathcal {L}}f(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(s)={\mathcal {L}}f(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/008d7a4a9897236cdb814000b8bf7d6146cc6fdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.521ex; height:2.843ex;" alt="{\displaystyle F(s)={\mathcal {L}}f(s)}"> be its Laplace transform. Then, within the region of convergence, we have <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\sigma +i\tau )=\int _{0}^{\infty }f(t)e^{-\sigma t}e^{-i\tau t}\,dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>+</mo> <mi>i</mi> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>σ</mi> <mi>t</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>τ</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\sigma +i\tau )=\int _{0}^{\infty }f(t)e^{-\sigma t}e^{-i\tau t}\,dt,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed583580cb790252b83166445e4a38ac8e373038" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.3ex; height:5.843ex;" alt="{\displaystyle F(\sigma +i\tau )=\int _{0}^{\infty }f(t)e^{-\sigma t}e^{-i\tau t}\,dt,}"></dd></dl> which is the Fourier transform of the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)e^{-\sigma t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>σ</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)e^{-\sigma t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92097a9015f5673497215e329fff088dec9e0de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.056ex; height:3.009ex;" alt="{\displaystyle f(t)e^{-\sigma t}}">.<a href="#cite_note-28">[28]</a> Indeed, the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> is a special case (under certain conditions) of the bilateral Laplace transform. The main difference is that the Fourier transform of a function is a complex function of a real variable (frequency), the Laplace transform of a function is a complex function of a complex variable. The Laplace transform is usually restricted to transformation of functions of t with t ≥ 0. A consequence of this restriction is that the Laplace transform of a function is a <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic function</a> of the variable s. Unlike the Fourier transform, the Laplace transform of a <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distribution</a> is generally a <a href="/wiki/Well-behaved" class="mw-redirect" title="Well-behaved">well-behaved</a> function. Techniques of complex variables can also be used to directly study Laplace transforms. As a holomorphic function, the Laplace transform has a <a href="/wiki/Power_series" title="Power series">power series</a> representation. This power series expresses a function as a linear superposition of <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moments</a> of the function. This perspective has applications in probability theory. Formally, the Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary argument s = iω<a href="#cite_note-29">[29]</a> <a href="#cite_note-30">[30]</a> when the condition explained below is fulfilled, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\hat {f}}(\omega )&={\mathcal {F}}\{f(t)\}\\[4pt]&={\mathcal {L}}\{f(t)\}|_{s=i\omega }=F(s)|_{s=i\omega }\\[4pt]&=\int _{-\infty }^{\infty }e^{-i\omega t}f(t)\,dt~.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mi>i</mi> <mi>ω</mi> </mrow> </msub> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mi>i</mi> <mi>ω</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mtext> </mtext> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\hat {f}}(\omega )&={\mathcal {F}}\{f(t)\}\\[4pt]&={\mathcal {L}}\{f(t)\}|_{s=i\omega }=F(s)|_{s=i\omega }\\[4pt]&=\int _{-\infty }^{\infty }e^{-i\omega t}f(t)\,dt~.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44fd8f6a7374f8e3ae9b00c7ca0107108784f9cd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.484ex; margin-bottom: -0.187ex; width:33.436ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}{\hat {f}}(\omega )&={\mathcal {F}}\{f(t)\}\\[4pt]&={\mathcal {L}}\{f(t)\}|_{s=i\omega }=F(s)|_{s=i\omega }\\[4pt]&=\int _{-\infty }^{\infty }e^{-i\omega t}f(t)\,dt~.\end{aligned}}}"> This convention of the Fourier transform (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}_{3}(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}_{3}(\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fca7249d253e4ed41cd847cdf257afce2f4b2dcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.008ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}_{3}(\omega )}"> in <a href="/wiki/Fourier_transform#Other_conventions" title="Fourier transform">Fourier transform § Other conventions</a>) requires a factor of <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠1/2π⁠ on the inverse Fourier transform. This relationship between the Laplace and Fourier transforms is often used to determine the <a href="/wiki/Frequency_spectrum" class="mw-redirect" title="Frequency spectrum">frequency spectrum</a> of a <a href="/wiki/Signal_(information_theory)" class="mw-redirect" title="Signal (information theory)">signal</a> or dynamical system. The above relation is valid as stated <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> the region of convergence (ROC) of F(s) contains the imaginary axis, σ = 0. For example, the function f(t) = cos(ω0t) has a Laplace transform F(s) = s/(s2 + ω02) whose ROC is Re(s) > 0. As s = iω0 is a pole of F(s), substituting s = iω in F(s) does not yield the Fourier transform of f(t)u(t), which contains terms proportional to the <a href="/wiki/Dirac_delta_functions" class="mw-redirect" title="Dirac delta functions">Dirac delta functions</a> δ(ω ± ω0). However, a relation of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{\sigma \to 0^{+}}F(\sigma +i\omega )={\hat {f}}(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <mi>F</mi> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>+</mo> <mi>i</mi> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{\sigma \to 0^{+}}F(\sigma +i\omega )={\hat {f}}(\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8af581ee3a0060f86b9b837a7ec677b18d5c77d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.016ex; height:4.843ex;" alt="{\displaystyle \lim _{\sigma \to 0^{+}}F(\sigma +i\omega )={\hat {f}}(\omega )}"> holds under much weaker conditions. For instance, this holds for the above example provided that the limit is understood as a <a href="/wiki/Weak_limit" class="mw-redirect" title="Weak limit">weak limit</a> of measures (see <a href="/wiki/Vague_topology" title="Vague topology">vague topology</a>). General conditions relating the limit of the Laplace transform of a function on the boundary to the Fourier transform take the form of <a href="/wiki/Paley%E2%80%93Wiener_theorem" title="Paley–Wiener theorem">Paley–Wiener theorems</a>. <div class="mw-heading mw-heading3"><h3 id="Mellin_transform">Mellin transform</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=16" title="Edit section: Mellin transform">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Mellin_transform" title="Mellin transform">Mellin transform</a></div> The Mellin transform and its inverse are related to the two-sided Laplace transform by a simple change of variables. If in the Mellin transform <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(s)={\mathcal {M}}\{g(\theta )\}=\int _{0}^{\infty }\theta ^{s}g(\theta )\,{\frac {d\theta }{\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> <mi>θ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(s)={\mathcal {M}}\{g(\theta )\}=\int _{0}^{\infty }\theta ^{s}g(\theta )\,{\frac {d\theta }{\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba07ffb3d7cbb20af402742490cd71c6b572c30a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.42ex; height:5.843ex;" alt="{\displaystyle G(s)={\mathcal {M}}\{g(\theta )\}=\int _{0}^{\infty }\theta ^{s}g(\theta )\,{\frac {d\theta }{\theta }}}"> we set θ = e−t we get a two-sided Laplace transform. <div class="mw-heading mw-heading3"><h3 id="Z-transform">Z-transform</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=17" title="Edit section: Z-transform">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Z-transform#Relationship_to_Laplace_transform" title="Z-transform">Z-transform § Relationship to Laplace transform</a></div> The unilateral or one-sided Z-transform is simply the Laplace transform of an ideally sampled signal with the substitution of <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z{\stackrel {\mathrm {def} }{{}={}}}e^{sT},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>T</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z{\stackrel {\mathrm {def} }{{}={}}}e^{sT},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd3f0fc4b7d863b35461b739efcd9b46aacedad6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.077ex; height:3.676ex;" alt="{\displaystyle z{\stackrel {\mathrm {def} }{{}={}}}e^{sT},}"> where T = 1/fs is the <a href="/wiki/Sampling_interval" class="mw-redirect" title="Sampling interval">sampling interval</a> (in units of time e.g., seconds) and fs is the <a href="/wiki/Sampling_rate" class="mw-redirect" title="Sampling rate">sampling rate</a> (in <a href="/wiki/Samples_per_second" class="mw-redirect" title="Samples per second">samples per second</a> or <a href="/wiki/Hertz" title="Hertz">hertz</a>). Let <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{T}(t)\ {\stackrel {\mathrm {def} }{=}}\ \sum _{n=0}^{\infty }\delta (t-nT)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{T}(t)\ {\stackrel {\mathrm {def} }{=}}\ \sum _{n=0}^{\infty }\delta (t-nT)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/511550196d09cdaa497d7f2b5ac9e106aceb2d0c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.089ex; height:6.843ex;" alt="{\displaystyle \Delta _{T}(t)\ {\stackrel {\mathrm {def} }{=}}\ \sum _{n=0}^{\infty }\delta (t-nT)}"> be a sampling impulse train (also called a <a href="/wiki/Dirac_comb" title="Dirac comb">Dirac comb</a>) and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x_{q}(t)&{\stackrel {\mathrm {def} }{{}={}}}x(t)\Delta _{T}(t)=x(t)\sum _{n=0}^{\infty }\delta (t-nT)\\&=\sum _{n=0}^{\infty }x(nT)\delta (t-nT)=\sum _{n=0}^{\infty }x[n]\delta (t-nT)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>x</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{q}(t)&{\stackrel {\mathrm {def} }{{}={}}}x(t)\Delta _{T}(t)=x(t)\sum _{n=0}^{\infty }\delta (t-nT)\\&=\sum _{n=0}^{\infty }x(nT)\delta (t-nT)=\sum _{n=0}^{\infty }x[n]\delta (t-nT)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c334ec8f4011a84a4a305f1ac7894fb6e1038cf0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:48.726ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}x_{q}(t)&{\stackrel {\mathrm {def} }{{}={}}}x(t)\Delta _{T}(t)=x(t)\sum _{n=0}^{\infty }\delta (t-nT)\\&=\sum _{n=0}^{\infty }x(nT)\delta (t-nT)=\sum _{n=0}^{\infty }x[n]\delta (t-nT)\end{aligned}}}"> be the sampled representation of the continuous-time x(t) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x[n]{\stackrel {\mathrm {def} }{{}={}}}x(nT)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x[n]{\stackrel {\mathrm {def} }{{}={}}}x(nT)~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b6eaf257e418b217759fe3b9b982993004af01" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.514ex; height:3.843ex;" alt="{\displaystyle x[n]{\stackrel {\mathrm {def} }{{}={}}}x(nT)~.}"> The Laplace transform of the sampled signal xq(t) is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}X_{q}(s)&=\int _{0^{-}}^{\infty }x_{q}(t)e^{-st}\,dt\\&=\int _{0^{-}}^{\infty }\sum _{n=0}^{\infty }x[n]\delta (t-nT)e^{-st}\,dt\\&=\sum _{n=0}^{\infty }x[n]\int _{0^{-}}^{\infty }\delta (t-nT)e^{-st}\,dt\\&=\sum _{n=0}^{\infty }x[n]e^{-nsT}~.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> <mi>s</mi> <mi>T</mi> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}X_{q}(s)&=\int _{0^{-}}^{\infty }x_{q}(t)e^{-st}\,dt\\&=\int _{0^{-}}^{\infty }\sum _{n=0}^{\infty }x[n]\delta (t-nT)e^{-st}\,dt\\&=\sum _{n=0}^{\infty }x[n]\int _{0^{-}}^{\infty }\delta (t-nT)e^{-st}\,dt\\&=\sum _{n=0}^{\infty }x[n]e^{-nsT}~.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee1fc990806a1d664c96eb253f19ff2def0f986" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.005ex; width:38.507ex; height:27.176ex;" alt="{\displaystyle {\begin{aligned}X_{q}(s)&=\int _{0^{-}}^{\infty }x_{q}(t)e^{-st}\,dt\\&=\int _{0^{-}}^{\infty }\sum _{n=0}^{\infty }x[n]\delta (t-nT)e^{-st}\,dt\\&=\sum _{n=0}^{\infty }x[n]\int _{0^{-}}^{\infty }\delta (t-nT)e^{-st}\,dt\\&=\sum _{n=0}^{\infty }x[n]e^{-nsT}~.\end{aligned}}}"> This is the precise definition of the unilateral Z-transform of the discrete function x[n] <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(z)=\sum _{n=0}^{\infty }x[n]z^{-n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(z)=\sum _{n=0}^{\infty }x[n]z^{-n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf7e6d1a9bf26a3017c79040c80295cb1f2eaa1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.323ex; height:6.843ex;" alt="{\displaystyle X(z)=\sum _{n=0}^{\infty }x[n]z^{-n}}"> with the substitution of z → esT. Comparing the last two equations, we find the relationship between the unilateral Z-transform and the Laplace transform of the sampled signal, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{q}(s)=X(z){\Big |}_{z=e^{sT}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>T</mi> </mrow> </msup> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{q}(s)=X(z){\Big |}_{z=e^{sT}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8c23e0ff230e75a3d72a8712ebdffa0e91f5fbb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:19.858ex; height:4.343ex;" alt="{\displaystyle X_{q}(s)=X(z){\Big |}_{z=e^{sT}}.}"> The similarity between the Z- and Laplace transforms is expanded upon in the theory of <a href="/wiki/Time_scale_calculus" class="mw-redirect" title="Time scale calculus">time scale calculus</a>. <div class="mw-heading mw-heading3"><h3 id="Borel_transform">Borel transform</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=18" title="Edit section: Borel transform">edit</a>]</div> The integral form of the <a href="/wiki/Borel_summation" title="Borel summation">Borel transform</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)=\int _{0}^{\infty }f(z)e^{-sz}\,dz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>z</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(s)=\int _{0}^{\infty }f(z)e^{-sz}\,dz}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0004fadac103e9953add733138e5211444dde29e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.468ex; height:5.843ex;" alt="{\displaystyle F(s)=\int _{0}^{\infty }f(z)e^{-sz}\,dz}"> is a special case of the Laplace transform for f an <a href="/wiki/Entire_function" title="Entire function">entire function</a> of exponential type, meaning that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(z)|\leq Ae^{B|z|}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>A</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(z)|\leq Ae^{B|z|}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/044af052128fde06ddcc3e64a66299af65a67878" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.559ex; height:3.343ex;" alt="{\displaystyle |f(z)|\leq Ae^{B|z|}}"> for some constants A and B. The generalized Borel transform allows a different weighting function to be used, rather than the exponential function, to transform functions not of exponential type. <a href="/wiki/Nachbin%27s_theorem" title="Nachbin's theorem">Nachbin's theorem</a> gives necessary and sufficient conditions for the Borel transform to be well defined. <div class="mw-heading mw-heading3"><h3 id="Fundamental_relationships">Fundamental relationships</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=19" title="Edit section: Fundamental relationships">edit</a>]</div> Since an ordinary Laplace transform can be written as a special case of a two-sided transform, and since the two-sided transform can be written as the sum of two one-sided transforms, the theory of the Laplace-, Fourier-, Mellin-, and Z-transforms are at bottom the same subject. However, a different point of view and different characteristic problems are associated with each of these four major integral transforms. <div class="mw-heading mw-heading2"><h2 id="Table_of_selected_Laplace_transforms">Table of selected Laplace transforms</h2>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=20" title="Edit section: Table of selected Laplace transforms">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/List_of_Laplace_transforms" title="List of Laplace transforms">List of Laplace transforms</a></div> The following table provides Laplace transforms for many common functions of a single variable.<a href="#cite_note-31">[31]</a><a href="#cite_note-32">[32]</a> For definitions and explanations, see the Explanatory Notes at the end of the table. Because the Laplace transform is a linear operator, <ul><li>The Laplace transform of a sum is the sum of Laplace transforms of each term.<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}\{f(t)+g(t)\}={\mathcal {L}}\{f(t)\}+{\mathcal {L}}\{g(t)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\{f(t)+g(t)\}={\mathcal {L}}\{f(t)\}+{\mathcal {L}}\{g(t)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4428a2eb9346d5896b51f85611c6e3daad1db9bb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.95ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}\{f(t)+g(t)\}={\mathcal {L}}\{f(t)\}+{\mathcal {L}}\{g(t)\}}"></li> <li>The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}\{af(t)\}=a{\mathcal {L}}\{f(t)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\{af(t)\}=a{\mathcal {L}}\{f(t)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/450c648b6dce57d9cb39dda8722afb1d47b3a115" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.27ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}\{af(t)\}=a{\mathcal {L}}\{f(t)\}}"></li></ul> Using this linearity, and various <a href="/wiki/List_of_trigonometric_identities" title="List of trigonometric identities">trigonometric</a>, <a href="/wiki/Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">hyperbolic</a>, and complex number (etc.) properties and/or identities, some Laplace transforms can be obtained from others more quickly than by using the definition directly. The unilateral Laplace transform takes as input a function whose time domain is the <a href="/wiki/Non-negative" class="mw-redirect" title="Non-negative">non-negative</a> reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, u(t). The entries of the table that involve a time delay τ are required to be <a href="/wiki/Causal_system" title="Causal system">causal</a> (meaning that τ > 0). A causal system is a system where the <a href="/wiki/Impulse_response" title="Impulse response">impulse response</a> h(t) is zero for all time t prior to t = 0. In general, the region of convergence for causal systems is not the same as that of <a href="/wiki/Anticausal_system" title="Anticausal system">anticausal systems</a>. <table class="wikitable" style="text-align: center;"> <caption>Selected Laplace transforms </caption> <tbody><tr> <th scope="col">Function </th> <th scope="col">Time domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)={\mathcal {L}}^{-1}\{F(s)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">{</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)={\mathcal {L}}^{-1}\{F(s)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb893031822346a930afd76491fc9a2d0fd48707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.928ex; height:3.176ex;" alt="{\displaystyle f(t)={\mathcal {L}}^{-1}\{F(s)\}}"> </th> <th scope="col">Laplace s-domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)={\mathcal {L}}\{f(t)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(s)={\mathcal {L}}\{f(t)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd0b873c7673d6ef7f6e9ea4e0e121d5771981f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.595ex; height:2.843ex;" alt="{\displaystyle F(s)={\mathcal {L}}\{f(t)\}}"> </th> <th scope="col">Region of convergence </th> <th scope="col">Reference </th></tr> <tr> <th scope="row">unit impulse </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (t)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da851d2e4afb359a36c5cb1c0dcfac6ed2b9ed03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.278ex; height:2.843ex;" alt="{\displaystyle \delta (t)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> </td> <td>all s </td> <td>inspection </td></tr> <tr> <th scope="row">delayed impulse </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (t-\tau )\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (t-\tau )\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d6cd3580d0ca4c10c5f4e119b7d59f648f2f7fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.321ex; height:2.843ex;" alt="{\displaystyle \delta (t-\tau )\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\tau s}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>τ</mi> <mi>s</mi> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-\tau s}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c29d0ab60e43bc00fa3dedeb8147126694ebd96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.796ex; height:2.509ex;" alt="{\displaystyle e^{-\tau s}\ }"> </td> <td>all s </td> <td>time shift of unit impulse </td></tr> <tr> <th scope="row">unit step </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(t)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb97101ef1486e114dacda61c4a5a775e3a2548" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.559ex; height:2.843ex;" alt="{\displaystyle u(t)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1 \over s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {1 \over s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e12052432638c711c0441bdd33cf609b8748ea7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {1 \over s}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0992e3b29be0b1bb2feec2e4682f43fe61c38b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.904ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>0}"> </td> <td>integrate unit impulse </td></tr> <tr> <th scope="row">delayed unit step </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(t-\tau )\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(t-\tau )\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f01928b97564610dbb5708a3eb426466d4718052" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.602ex; height:2.843ex;" alt="{\displaystyle u(t-\tau )\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{s}}e^{-\tau s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>τ</mi> <mi>s</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{s}}e^{-\tau s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3929545c83c9b542c44d456bc1987b0a1f1dc774" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.214ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{s}}e^{-\tau s}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0992e3b29be0b1bb2feec2e4682f43fe61c38b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.904ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>0}"> </td> <td>time shift of unit step </td></tr> <tr> <th scope="row">product of delayed function and delayed step </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t-\tau )u(t-\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t-\tau )u(t-\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/578588d732f29a9be348f9a2d465e2e4e8bd73ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.991ex; height:2.843ex;" alt="{\displaystyle f(t-\tau )u(t-\tau )}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-s\tau }{\mathcal {L}}\{f(t)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>τ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-s\tau }{\mathcal {L}}\{f(t)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05071ec6c957ba4fa563d5e2d758e62bda991995" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.071ex; height:3.009ex;" alt="{\displaystyle e^{-s\tau }{\mathcal {L}}\{f(t)\}}"> </td> <td> </td> <td>u-substitution, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=t-\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=t-\tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9f136ba801f6bd00ee985e98cec37235490289b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.31ex; height:2.176ex;" alt="{\displaystyle u=t-\tau }"> </td></tr> <tr> <th>rectangular impulse </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(t)-u(t-\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(t)-u(t-\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57b648902f871bd93e0df4ba915ee55372b97bb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.84ex; height:2.843ex;" alt="{\displaystyle u(t)-u(t-\tau )}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{s}}(1-e^{-\tau s})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>τ</mi> <mi>s</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{s}}(1-e^{-\tau s})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02f84c63c3275bcdda87a430c15511e203486c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.026ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{s}}(1-e^{-\tau s})}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0992e3b29be0b1bb2feec2e4682f43fe61c38b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.904ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>0}"> </td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Ramp_function" title="Ramp function">ramp</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\cdot u(t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\cdot u(t)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb3cde9545296c49fba514af248a342570095492" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.078ex; height:2.843ex;" alt="{\displaystyle t\cdot u(t)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{s^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{s^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46948af3cd3b6074ff83a651e87fa2414c7f0f5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:2.981ex; height:5.509ex;" alt="{\displaystyle {\frac {1}{s^{2}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0992e3b29be0b1bb2feec2e4682f43fe61c38b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.904ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>0}"> </td> <td>integrate unit impulse twice </td></tr> <tr> <th scope="row">nth power (for integer n) </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t^{n}\cdot u(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t^{n}\cdot u(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5b896e253b06b1f04b53713f84c78c09af74433" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.716ex; height:2.843ex;" alt="{\displaystyle t^{n}\cdot u(t)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n! \over s^{n+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n! \over s^{n+1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea1fad01d457183aab6c33d2afd840e118551ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:5.246ex; height:5.676ex;" alt="{\displaystyle {n! \over s^{n+1}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0992e3b29be0b1bb2feec2e4682f43fe61c38b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.904ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>0}"> (n > −1) </td> <td>integrate unit step n times </td></tr> <tr> <th scope="row">qth power (for complex q) </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t^{q}\cdot u(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t^{q}\cdot u(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b38aae75d1d01f8dcd86fd38ff201f9de2e7bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.486ex; height:2.843ex;" alt="{\displaystyle t^{q}\cdot u(t)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\operatorname {\Gamma } (q+1) \over s^{q+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">Γ</mi> </mrow> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\operatorname {\Gamma } (q+1) \over s^{q+1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff5da3f715028ed58491ffda69c43cf66000950c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.171ex; height:6.009ex;" alt="{\displaystyle {\operatorname {\Gamma } (q+1) \over s^{q+1}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0992e3b29be0b1bb2feec2e4682f43fe61c38b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.904ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>0}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (q)>-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>></mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (q)>-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7a76f768d20256f5e6a4ca9c1752a4a51639514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.691ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (q)>-1}"> </td> <td><a href="#cite_note-33">[33]</a><a href="#cite_note-34">[34]</a> </td></tr> <tr> <th scope="row">nth root </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{t}}\cdot u(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{t}}\cdot u(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4345a4c33a88daeb8ec5a3002d02d62f66ff3fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.433ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{t}}\cdot u(t)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1 \over s^{{\frac {1}{n}}+1}}\operatorname {\Gamma } \left({\frac {1}{n}}+1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">Γ</mi> </mrow> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {1 \over s^{{\frac {1}{n}}+1}}\operatorname {\Gamma } \left({\frac {1}{n}}+1\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72c19f00d756dcde3465b231a14e82a22a7fd61c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.391ex; height:6.676ex;" alt="{\displaystyle {1 \over s^{{\frac {1}{n}}+1}}\operatorname {\Gamma } \left({\frac {1}{n}}+1\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0992e3b29be0b1bb2feec2e4682f43fe61c38b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.904ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>0}"> </td> <td>Set q = 1/n above. </td></tr> <tr> <th scope="row">nth power with frequency shift </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t^{n}e^{-\alpha t}\cdot u(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>t</mi> </mrow> </msup> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t^{n}e^{-\alpha t}\cdot u(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ed5279bfe16ff5fcb23fd2fbdbea2f07903b4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.956ex; height:3.009ex;" alt="{\displaystyle t^{n}e^{-\alpha t}\cdot u(t)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n!}{(s+\alpha )^{n+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>α</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n!}{(s+\alpha )^{n+1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ec193d9657b8fd89deb9893c4ccc6ad18746a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.383ex; height:6.176ex;" alt="{\displaystyle {\frac {n!}{(s+\alpha )^{n+1}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>-\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mo>−</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>-\alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2a74e2a87df4f4d30a2aa6c10c95cc701d211d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.037ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>-\alpha }"> </td> <td>Integrate unit step, apply frequency shift </td></tr> <tr> <th scope="row">delayed nth power with frequency shift </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t-\tau )^{n}e^{-\alpha (t-\tau )}\cdot u(t-\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t-\tau )^{n}e^{-\alpha (t-\tau )}\cdot u(t-\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3605609561a79361679ca30ff80af575a370a902" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.257ex; height:3.343ex;" alt="{\displaystyle (t-\tau )^{n}e^{-\alpha (t-\tau )}\cdot u(t-\tau )}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n!\cdot e^{-\tau s}}{(s+\alpha )^{n+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>τ</mi> <mi>s</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>α</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n!\cdot e^{-\tau s}}{(s+\alpha )^{n+1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed9471e996bb67e33f1a8296e375844af1c79505" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.383ex; height:6.343ex;" alt="{\displaystyle {\frac {n!\cdot e^{-\tau s}}{(s+\alpha )^{n+1}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>-\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mo>−</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>-\alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2a74e2a87df4f4d30a2aa6c10c95cc701d211d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.037ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>-\alpha }"> </td> <td>integrate unit step, apply frequency shift, apply time shift </td></tr> <tr> <th scope="row"><a href="/wiki/Exponential_decay" title="Exponential decay">exponential decay</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\alpha t}\cdot u(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>t</mi> </mrow> </msup> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-\alpha t}\cdot u(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98695eb1dc70cd1151643626ec64661f80005394" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.898ex; height:3.009ex;" alt="{\displaystyle e^{-\alpha t}\cdot u(t)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1 \over s+\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>+</mo> <mi>α</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {1 \over s+\alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0229f76cec6c45e76f727146329996a4cd9e895" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.255ex; height:5.343ex;" alt="{\displaystyle {1 \over s+\alpha }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>-\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mo>−</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>-\alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2a74e2a87df4f4d30a2aa6c10c95cc701d211d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.037ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>-\alpha }"> </td> <td>Frequency shift of unit step </td></tr> <tr> <th scope="row"><a href="/wiki/Two-sided_Laplace_transform" title="Two-sided Laplace transform">two-sided</a> exponential decay (only for bilateral transform) </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\alpha |t|}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-\alpha |t|}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9ac4a67cfcbc97320effc38c9d84cdad43d8e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.735ex; height:2.843ex;" alt="{\displaystyle e^{-\alpha |t|}\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {2\alpha \over \alpha ^{2}-s^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>α</mi> </mrow> <mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {2\alpha \over \alpha ^{2}-s^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0051efbbf8f1bd6df33b217d856e06fc7a89b491" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.363ex; height:5.676ex;" alt="{\displaystyle {2\alpha \over \alpha ^{2}-s^{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\alpha <\operatorname {Re} (s)<\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>α</mi> <mo><</mo> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\alpha <\operatorname {Re} (s)<\alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c160ff756a28749d751a2795c0b1d4b03379b0f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.623ex; height:2.843ex;" alt="{\displaystyle -\alpha <\operatorname {Re} (s)<\alpha }"> </td> <td>Frequency shift of unit step </td></tr> <tr> <th scope="row">exponential approach </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-e^{-\alpha t})\cdot u(t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>t</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-e^{-\alpha t})\cdot u(t)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c701de6d8a894649dd3513cc3e97844172b903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.29ex; height:3.009ex;" alt="{\displaystyle (1-e^{-\alpha t})\cdot u(t)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha }{s(s+\alpha )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha }{s(s+\alpha )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05505ff80466f8350293e9e58644838f4192da8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.154ex; height:5.509ex;" alt="{\displaystyle {\frac {\alpha }{s(s+\alpha )}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0992e3b29be0b1bb2feec2e4682f43fe61c38b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.904ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>0}"> </td> <td>unit step minus exponential decay </td></tr> <tr> <th scope="row"><a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\omega t)\cdot u(t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\omega t)\cdot u(t)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02dde40ceb574c28a10539f224e87a29e7d63648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.189ex; height:2.843ex;" alt="{\displaystyle \sin(\omega t)\cdot u(t)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\omega \over s^{2}+\omega ^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\omega \over s^{2}+\omega ^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/876ae003c68ad9023b3f0e85c4f93d6dbc1e0eaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.321ex; height:5.176ex;" alt="{\displaystyle {\omega \over s^{2}+\omega ^{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0992e3b29be0b1bb2feec2e4682f43fe61c38b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.904ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>0}"> </td> <td><a href="#cite_note-FOOTNOTEBracewell1978227-35">[35]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\omega t)\cdot u(t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\omega t)\cdot u(t)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d975de98dca5bf39d786f8660da37e2dbfdc8292" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.444ex; height:2.843ex;" alt="{\displaystyle \cos(\omega t)\cdot u(t)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {s \over s^{2}+\omega ^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {s \over s^{2}+\omega ^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21481b6bd59e4250e5eac3503f8ee588dcb9c667" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.321ex; height:5.176ex;" alt="{\displaystyle {s \over s^{2}+\omega ^{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0992e3b29be0b1bb2feec2e4682f43fe61c38b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.904ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>0}"> </td> <td><a href="#cite_note-FOOTNOTEBracewell1978227-35">[35]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Hyperbolic_sine" class="mw-redirect" title="Hyperbolic sine">hyperbolic sine</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sinh(\alpha t)\cdot u(t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sinh(\alpha t)\cdot u(t)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7718b00ca36d25104361aeab18cf8e1e7f9a627" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.523ex; height:2.843ex;" alt="{\displaystyle \sinh(\alpha t)\cdot u(t)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\alpha \over s^{2}-\alpha ^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\alpha \over s^{2}-\alpha ^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e6ea24afce8149199cb1f61d500440b0808e79e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.363ex; height:5.176ex;" alt="{\displaystyle {\alpha \over s^{2}-\alpha ^{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>\left|\alpha \right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mrow> <mo>|</mo> <mi>α</mi> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>\left|\alpha \right|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c777547ea8a3b6b843e45f9601b32221a75610cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.522ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>\left|\alpha \right|}"> </td> <td><a href="#cite_note-FOOTNOTEWilliams197388-36">[36]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Hyperbolic_cosine" class="mw-redirect" title="Hyperbolic cosine">hyperbolic cosine</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cosh(\alpha t)\cdot u(t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cosh(\alpha t)\cdot u(t)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b39113c0f6e3efd283ccfda7060de3123c82888" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.779ex; height:2.843ex;" alt="{\displaystyle \cosh(\alpha t)\cdot u(t)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {s \over s^{2}-\alpha ^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {s \over s^{2}-\alpha ^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c80ab2a34d0f9ff049027a729381874e9bd830c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.363ex; height:5.176ex;" alt="{\displaystyle {s \over s^{2}-\alpha ^{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>\left|\alpha \right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mrow> <mo>|</mo> <mi>α</mi> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>\left|\alpha \right|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c777547ea8a3b6b843e45f9601b32221a75610cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.522ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>\left|\alpha \right|}"> </td> <td><a href="#cite_note-FOOTNOTEWilliams197388-36">[36]</a> </td></tr> <tr> <th scope="row">exponentially decaying sine wave </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\alpha t}\sin(\omega t)\cdot u(t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>t</mi> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-\alpha t}\sin(\omega t)\cdot u(t)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5def66018fbfc9d9d61f875183e7e33240fdd8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.816ex; height:3.009ex;" alt="{\displaystyle e^{-\alpha t}\sin(\omega t)\cdot u(t)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\omega \over (s+\alpha )^{2}+\omega ^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>α</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\omega \over (s+\alpha )^{2}+\omega ^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55f4baa256fac73b43492a29cab9d49d87611d5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.459ex; height:5.509ex;" alt="{\displaystyle {\omega \over (s+\alpha )^{2}+\omega ^{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>-\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mo>−</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>-\alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2a74e2a87df4f4d30a2aa6c10c95cc701d211d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.037ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>-\alpha }"> </td> <td><a href="#cite_note-FOOTNOTEBracewell1978227-35">[35]</a> </td></tr> <tr> <th scope="row">exponentially decaying cosine wave </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\alpha t}\cos(\omega t)\cdot u(t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>t</mi> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-\alpha t}\cos(\omega t)\cdot u(t)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a05cfb92da9b3d2d19165a71df2f0f080ce2a44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.071ex; height:3.009ex;" alt="{\displaystyle e^{-\alpha t}\cos(\omega t)\cdot u(t)\ }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {s+\alpha \over (s+\alpha )^{2}+\omega ^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mo>+</mo> <mi>α</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>α</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {s+\alpha \over (s+\alpha )^{2}+\omega ^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c63c48f07a46bcc5b39f1d87c0ea09403c1561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.459ex; height:5.843ex;" alt="{\displaystyle {s+\alpha \over (s+\alpha )^{2}+\omega ^{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>-\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mo>−</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>-\alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2a74e2a87df4f4d30a2aa6c10c95cc701d211d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.037ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>-\alpha }"> </td> <td><a href="#cite_note-FOOTNOTEBracewell1978227-35">[35]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(t)\cdot u(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(t)\cdot u(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb3b8f4fbbeafadc26da19ede9e3e10601d6f54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.246ex; height:2.843ex;" alt="{\displaystyle \ln(t)\cdot u(t)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{1 \over s}\left[\ln(s)+\gamma \right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>γ</mi> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{1 \over s}\left[\ln(s)+\gamma \right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11e5254f2039e5373fac5178eb8da59890d0979f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.429ex; height:5.176ex;" alt="{\displaystyle -{1 \over s}\left[\ln(s)+\gamma \right]}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0992e3b29be0b1bb2feec2e4682f43fe61c38b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.904ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>0}"> </td> <td><a href="#cite_note-FOOTNOTEWilliams197388-36">[36]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Bessel_function" title="Bessel function">Bessel function</a> of the first kind, of order n </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{n}(\omega t)\cdot u(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{n}(\omega t)\cdot u(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a71a39dba1701407eadcc42453bf821e0705a3b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.261ex; height:2.843ex;" alt="{\displaystyle J_{n}(\omega t)\cdot u(t)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\left({\sqrt {s^{2}+\omega ^{2}}}-s\right)^{\!n}}{\omega ^{n}{\sqrt {s^{2}+\omega ^{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>−</mo> <mi>s</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace" /> <mi>n</mi> </mrow> </msup> <mrow> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\left({\sqrt {s^{2}+\omega ^{2}}}-s\right)^{\!n}}{\omega ^{n}{\sqrt {s^{2}+\omega ^{2}}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a15fc6e827c36a9c69326707fea8fb3133261838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.183ex; height:8.843ex;" alt="{\displaystyle {\frac {\left({\sqrt {s^{2}+\omega ^{2}}}-s\right)^{\!n}}{\omega ^{n}{\sqrt {s^{2}+\omega ^{2}}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0992e3b29be0b1bb2feec2e4682f43fe61c38b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.904ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>0}"> (n > −1) </td> <td><a href="#cite_note-FOOTNOTEWilliams197389-37">[37]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Error_function" title="Error function">Error function</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erf} (t)\cdot u(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>erf</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (t)\cdot u(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbaab4b629505dd625c90944aba1981699dad121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.962ex; height:2.843ex;" alt="{\displaystyle \operatorname {erf} (t)\cdot u(t)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{s}}e^{(1/4)s^{2}}\!\left(1-\operatorname {erf} {\frac {s}{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo stretchy="false">)</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>erf</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{s}}e^{(1/4)s^{2}}\!\left(1-\operatorname {erf} {\frac {s}{2}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bdd0bc2936c98c66536e9c5620eeb863bec0df4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.482ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{s}}e^{(1/4)s^{2}}\!\left(1-\operatorname {erf} {\frac {s}{2}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0992e3b29be0b1bb2feec2e4682f43fe61c38b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.904ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>0}"> </td> <td><a href="#cite_note-FOOTNOTEWilliams197389-37">[37]</a> </td></tr> <tr> <td colspan="5" style="text-align: left;">Explanatory notes: <style data-mw-deduplicate="TemplateStyles:r1216972533">.mw-parser-output .col-begin{border-collapse:collapse;padding:0;color:inherit;width:100%;border:0;margin:0}.mw-parser-output .col-begin-small{font-size:90%}.mw-parser-output .col-break{vertical-align:top;text-align:left}.mw-parser-output .col-break-2{width:50%}.mw-parser-output .col-break-3{width:33.3%}.mw-parser-output .col-break-4{width:25%}.mw-parser-output .col-break-5{width:20%}@media(max-width:720px){.mw-parser-output .col-begin,.mw-parser-output .col-begin>tbody,.mw-parser-output .col-begin>tbody>tr,.mw-parser-output .col-begin>tbody>tr>td{display:block!important;width:100%!important}.mw-parser-output .col-break{padding-left:0!important}}</style><div> <table class="col-begin" role="presentation"> <tbody><tr> <td class="col-break"> <ul><li>u(t) represents the <a href="/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a>.</li> <li>δ represents the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>.</li> <li>Γ(z) represents the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a>.</li> <li>γ is the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>.</li></ul> </td> <td class="col-break"> <ul><li>t, a real number, typically represents time, although it can represent any independent dimension.</li> <li>s is the <a href="/wiki/Complex_number" title="Complex number">complex</a> frequency domain parameter, and Re(s) is its <a href="/wiki/Real_part" class="mw-redirect" title="Real part">real part</a>.</li> <li>α, β, τ, and ω are <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>.</li> <li>n is an <a href="/wiki/Integer" title="Integer">integer</a>.</li></ul> </td></tr></tbody></table></div> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="s-domain_equivalent_circuits_and_impedances">s-domain equivalent circuits and impedances</h2>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=21" title="Edit section: s-domain equivalent circuits and impedances">edit</a>]</div> The Laplace transform is often used in <a href="/wiki/Network_analysis_(electrical_circuits)" title="Network analysis (electrical circuits)">circuit analysis</a>, and simple conversions to the s-domain of circuit elements can be made. Circuit elements can be transformed into <a href="/wiki/Electrical_impedance" title="Electrical impedance">impedances</a>, very similar to <a href="/wiki/Phasor_(sine_waves)" class="mw-redirect" title="Phasor (sine waves)">phasor</a> impedances. Here is a summary of equivalents: <dl><dd><figure class="mw-halign-center" typeof="mw:File/Frameless"><a href="/wiki/File:S-Domain_circuit_equivalents.svg" class="mw-file-description" title="s-domain equivalent circuits"><img alt="s-domain equivalent circuits" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/S-Domain_circuit_equivalents.svg/400px-S-Domain_circuit_equivalents.svg.png" decoding="async" width="400" height="274" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/S-Domain_circuit_equivalents.svg/600px-S-Domain_circuit_equivalents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/S-Domain_circuit_equivalents.svg/800px-S-Domain_circuit_equivalents.svg.png 2x" data-file-width="615" data-file-height="422" /></a><figcaption>s-domain equivalent circuits</figcaption></figure></dd></dl> Note that the resistor is exactly the same in the time domain and the s-domain. The sources are put in if there are initial conditions on the circuit elements. For example, if a capacitor has an initial voltage across it, or if the inductor has an initial current through it, the sources inserted in the s-domain account for that. The equivalents for current and voltage sources are simply derived from the transformations in the table above. <div class="mw-heading mw-heading2"><h2 id="Examples_and_applications">Examples and applications</h2>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=22" title="Edit section: Examples and applications">edit</a>]</div> The Laplace transform is used frequently in <a href="/wiki/Engineering" title="Engineering">engineering</a> and <a href="/wiki/Physics" title="Physics">physics</a>; the output of a <a href="/wiki/Linear_time-invariant_system" title="Linear time-invariant system">linear time-invariant system</a> can be calculated by convolving its unit impulse response with the input signal. Performing this calculation in Laplace space turns the convolution into a multiplication; the latter being easier to solve because of its algebraic form. For more information, see <a href="/wiki/Control_theory" title="Control theory">control theory</a>. The Laplace transform is invertible on a large class of functions. Given a simple mathematical or functional description of an input or output to a <a href="/wiki/System" title="System">system</a>, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.<a href="#cite_note-38">[38]</a> The Laplace transform can also be used to solve differential equations and is used extensively in <a href="/wiki/Mechanical_engineering" title="Mechanical engineering">mechanical engineering</a> and <a href="/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a>. The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra. The original differential equation can then be solved by applying the inverse Laplace transform. English electrical engineer <a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a> first proposed a similar scheme, although without using the Laplace transform; and the resulting operational calculus is credited as the Heaviside calculus. <div class="mw-heading mw-heading3"><h3 id="Evaluating_improper_integrals">Evaluating improper integrals</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=23" title="Edit section: Evaluating improper integrals">edit</a>]</div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}\left\{f(t)\right\}=F(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\left\{f(t)\right\}=F(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/457c90bbb581abb2c11aec90c91413a91c5499c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.982ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}\left\{f(t)\right\}=F(s)}">. Then (see the table above) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{s}{\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\partial _{s}\int _{0}^{\infty }{\frac {f(t)}{t}}e^{-st}\,dt=-\int _{0}^{\infty }f(t)e^{-st}dt=-F(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>t</mi> </mfrac> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>t</mi> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{s}{\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\partial _{s}\int _{0}^{\infty }{\frac {f(t)}{t}}e^{-st}\,dt=-\int _{0}^{\infty }f(t)e^{-st}dt=-F(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acea4575b485f1df65aa88013d1b472cd6ee8fc9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:63.603ex; height:6.343ex;" alt="{\displaystyle \partial _{s}{\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\partial _{s}\int _{0}^{\infty }{\frac {f(t)}{t}}e^{-st}\,dt=-\int _{0}^{\infty }f(t)e^{-st}dt=-F(s)}"> From which one gets: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\int _{s}^{\infty }F(p)\,dp.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>t</mi> </mfrac> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>F</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\int _{s}^{\infty }F(p)\,dp.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fb900bae03ad5146b2c65fdae139a7df6e649a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.205ex; height:6.343ex;" alt="{\displaystyle {\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\int _{s}^{\infty }F(p)\,dp.}"> In the limit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\rightarrow 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\rightarrow 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18707f5a3ad0fa29ff1f32ef409349f235576588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.867ex; height:2.176ex;" alt="{\displaystyle s\rightarrow 0}">, one gets <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }{\frac {f(t)}{t}}\,dt=\int _{0}^{\infty }F(p)\,dp,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>t</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>F</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }{\frac {f(t)}{t}}\,dt=\int _{0}^{\infty }F(p)\,dp,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fbe780ed0e9214ce7f97095ef53f109c3524073" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.897ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{\infty }{\frac {f(t)}{t}}\,dt=\int _{0}^{\infty }F(p)\,dp,}"> provided that the interchange of limits can be justified. This is often possible as a consequence of the <a href="/wiki/Final_value_theorem#Final_Value_Theorem_for_improperly_integrable_functions_(Abel's_theorem_for_integrals)" title="Final value theorem">final value theorem</a>. Even when the interchange cannot be justified the calculation can be suggestive. For example, with a ≠ 0 ≠ b, proceeding formally one has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int _{0}^{\infty }{\frac {\cos(at)-\cos(bt)}{t}}\,dt&=\int _{0}^{\infty }\left({\frac {p}{p^{2}+a^{2}}}-{\frac {p}{p^{2}+b^{2}}}\right)\,dp\\[6pt]&=\left[{\frac {1}{2}}\ln {\frac {p^{2}+a^{2}}{p^{2}+b^{2}}}\right]_{0}^{\infty }={\frac {1}{2}}\ln {\frac {b^{2}}{a^{2}}}=\ln \left|{\frac {b}{a}}\right|.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>t</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{0}^{\infty }{\frac {\cos(at)-\cos(bt)}{t}}\,dt&=\int _{0}^{\infty }\left({\frac {p}{p^{2}+a^{2}}}-{\frac {p}{p^{2}+b^{2}}}\right)\,dp\\[6pt]&=\left[{\frac {1}{2}}\ln {\frac {p^{2}+a^{2}}{p^{2}+b^{2}}}\right]_{0}^{\infty }={\frac {1}{2}}\ln {\frac {b^{2}}{a^{2}}}=\ln \left|{\frac {b}{a}}\right|.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/150eb829ed85ccdec437af079d262fc08428699c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:65.645ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}\int _{0}^{\infty }{\frac {\cos(at)-\cos(bt)}{t}}\,dt&=\int _{0}^{\infty }\left({\frac {p}{p^{2}+a^{2}}}-{\frac {p}{p^{2}+b^{2}}}\right)\,dp\\[6pt]&=\left[{\frac {1}{2}}\ln {\frac {p^{2}+a^{2}}{p^{2}+b^{2}}}\right]_{0}^{\infty }={\frac {1}{2}}\ln {\frac {b^{2}}{a^{2}}}=\ln \left|{\frac {b}{a}}\right|.\end{aligned}}}"> The validity of this identity can be proved by other means. It is an example of a <a href="/wiki/Frullani_integral" title="Frullani integral">Frullani integral</a>. Another example is <a href="/wiki/Dirichlet_integral" title="Dirichlet integral">Dirichlet integral</a>. <div class="mw-heading mw-heading3"><h3 id="Complex_impedance_of_a_capacitor">Complex impedance of a capacitor</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=24" title="Edit section: Complex impedance of a capacitor">edit</a>]</div> In the theory of <a href="/wiki/Electrical_circuit" class="mw-redirect" title="Electrical circuit">electrical circuits</a>, the current flow in a <a href="/wiki/Capacitor" title="Capacitor">capacitor</a> is proportional to the capacitance and rate of change in the electrical potential (with equations as for the <a href="/wiki/International_System_of_Units" title="International System of Units">SI</a> unit system). Symbolically, this is expressed by the differential equation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=C{dv \over dt},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>v</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=C{dv \over dt},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83bfad94907c3d3111a094ae9a2643abd7a7e437" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.494ex; height:5.509ex;" alt="{\displaystyle i=C{dv \over dt},}"> where C is the capacitance of the capacitor, i = i(t) is the <a href="/wiki/Electric_current" title="Electric current">electric current</a> through the capacitor as a function of time, and v = v(t) is the <a href="/wiki/Electrostatic_potential" class="mw-redirect" title="Electrostatic potential">voltage</a> across the terminals of the capacitor, also as a function of time. Taking the Laplace transform of this equation, we obtain <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(s)=C(sV(s)-V_{0}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(s)=C(sV(s)-V_{0}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a142940c134a6388521a7dee9692ee87c32fe30" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.42ex; height:2.843ex;" alt="{\displaystyle I(s)=C(sV(s)-V_{0}),}"> where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}I(s)&={\mathcal {L}}\{i(t)\},\\V(s)&={\mathcal {L}}\{v(t)\},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>I</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I(s)&={\mathcal {L}}\{i(t)\},\\V(s)&={\mathcal {L}}\{v(t)\},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f198d497ac10e65162028228912437b31fb313" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.889ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}I(s)&={\mathcal {L}}\{i(t)\},\\V(s)&={\mathcal {L}}\{v(t)\},\end{aligned}}}"> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}=v(0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{0}=v(0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf78b400ca74acedeedbda8cf3fb793103659ce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.254ex; height:2.843ex;" alt="{\displaystyle V_{0}=v(0).}"> Solving for V(s) we have <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(s)={I(s) \over sC}+{V_{0} \over s}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>s</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>s</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(s)={I(s) \over sC}+{V_{0} \over s}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f110c587519eb4a4d6ffabf1b58624b0ad97231" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.426ex; height:5.843ex;" alt="{\displaystyle V(s)={I(s) \over sC}+{V_{0} \over s}.}"> The definition of the complex impedance Z (in <a href="/wiki/Ohm" title="Ohm">ohms</a>) is the ratio of the complex voltage V divided by the complex current I while holding the initial state V0 at zero: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(s)=\left.{V(s) \over I(s)}\right|_{V_{0}=0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(s)=\left.{V(s) \over I(s)}\right|_{V_{0}=0}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6711003350f4559c32bca99676f9ae3e2f000df7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.618ex; height:6.843ex;" alt="{\displaystyle Z(s)=\left.{V(s) \over I(s)}\right|_{V_{0}=0}.}"> Using this definition and the previous equation, we find: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(s)={\frac {1}{sC}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(s)={\frac {1}{sC}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9aace1d29c498b7585d461970a874aa2ca5fefe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.018ex; height:5.343ex;" alt="{\displaystyle Z(s)={\frac {1}{sC}},}"> which is the correct expression for the complex impedance of a capacitor. In addition, the Laplace transform has large applications in control theory. <div class="mw-heading mw-heading3"><h3 id="Impulse_response">Impulse response</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=25" title="Edit section: Impulse response">edit</a>]</div> Consider a linear time-invariant system with <a href="/wiki/Transfer_function" title="Transfer function">transfer function</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(s)={\frac {1}{(s+\alpha )(s+\beta )}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(s)={\frac {1}{(s+\alpha )(s+\beta )}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/553a78b681373612c87e6d31f6a2b52080d1871b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.845ex; height:6.009ex;" alt="{\displaystyle H(s)={\frac {1}{(s+\alpha )(s+\beta )}}.}"> The <a href="/wiki/Impulse_response" title="Impulse response">impulse response</a> is simply the inverse Laplace transform of this transfer function: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(t)={\mathcal {L}}^{-1}\{H(s)\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">{</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(t)={\mathcal {L}}^{-1}\{H(s)\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c2fcfed3bfaf30f53ccc8c2b9d1064a4c51b9dc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.958ex; height:3.176ex;" alt="{\displaystyle h(t)={\mathcal {L}}^{-1}\{H(s)\}.}"> <dl><dt>Partial fraction expansion</dt></dl> To evaluate this inverse transform, we begin by expanding H(s) using the method of partial fraction expansion, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{(s+\alpha )(s+\beta )}}={P \over s+\alpha }+{R \over s+\beta }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <mrow> <mi>s</mi> <mo>+</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mrow> <mi>s</mi> <mo>+</mo> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{(s+\alpha )(s+\beta )}}={P \over s+\alpha }+{R \over s+\beta }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18e59114cab3e1242b577c78ce884fe1225b4871" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.075ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{(s+\alpha )(s+\beta )}}={P \over s+\alpha }+{R \over s+\beta }.}"> The unknown constants P and R are the <a href="/wiki/Residue_(complex_analysis)" title="Residue (complex analysis)">residues</a> located at the corresponding poles of the transfer function. Each residue represents the relative contribution of that <a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singularity</a> to the transfer function's overall shape. By the <a href="/wiki/Residue_theorem" title="Residue theorem">residue theorem</a>, the inverse Laplace transform depends only upon the poles and their residues. To find the residue P, we multiply both sides of the equation by s + α to get <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{s+\beta }}=P+{R(s+\alpha ) \over s+\beta }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>+</mo> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>P</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>s</mi> <mo>+</mo> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{s+\beta }}=P+{R(s+\alpha ) \over s+\beta }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da6d4af6ff6e108f5079733011d48d7fd47e35c5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.258ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{s+\beta }}=P+{R(s+\alpha ) \over s+\beta }.}"> Then by letting s = −α, the contribution from R vanishes and all that is left is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\left.{1 \over s+\beta }\right|_{s=-\alpha }={1 \over \beta -\alpha }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>+</mo> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mo>−</mo> <mi>α</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>β</mi> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\left.{1 \over s+\beta }\right|_{s=-\alpha }={1 \over \beta -\alpha }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/667e086ca22b4819f4188e6a30c83baa4fbb329d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.443ex; height:6.176ex;" alt="{\displaystyle P=\left.{1 \over s+\beta }\right|_{s=-\alpha }={1 \over \beta -\alpha }.}"> Similarly, the residue R is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=\left.{1 \over s+\alpha }\right|_{s=-\beta }={1 \over \alpha -\beta }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>+</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mo>−</mo> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>α</mi> <mo>−</mo> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\left.{1 \over s+\alpha }\right|_{s=-\beta }={1 \over \alpha -\beta }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09980f06092ea8dccbaa90eaf11ceb6cc7225b4f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.508ex; height:6.009ex;" alt="{\displaystyle R=\left.{1 \over s+\alpha }\right|_{s=-\beta }={1 \over \alpha -\beta }.}"> Note that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R={-1 \over \beta -\alpha }=-P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>β</mi> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={-1 \over \beta -\alpha }=-P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c637470458d06ed421d04e21d52a9add74e4bf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.011ex; height:5.676ex;" alt="{\displaystyle R={-1 \over \beta -\alpha }=-P}"> and so the substitution of R and P into the expanded expression for H(s) gives <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(s)=\left({\frac {1}{\beta -\alpha }}\right)\cdot \left({1 \over s+\alpha }-{1 \over s+\beta }\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>β</mi> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>+</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>+</mo> <mi>β</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(s)=\left({\frac {1}{\beta -\alpha }}\right)\cdot \left({1 \over s+\alpha }-{1 \over s+\beta }\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8526941d3c46c3c1430896441589019926caba3e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.307ex; height:6.176ex;" alt="{\displaystyle H(s)=\left({\frac {1}{\beta -\alpha }}\right)\cdot \left({1 \over s+\alpha }-{1 \over s+\beta }\right).}"> Finally, using the linearity property and the known transform for exponential decay (see Item #3 in the Table of Laplace Transforms, above), we can take the inverse Laplace transform of H(s) to obtain <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(t)={\mathcal {L}}^{-1}\{H(s)\}={\frac {1}{\beta -\alpha }}\left(e^{-\alpha t}-e^{-\beta t}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">{</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>β</mi> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>t</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mi>t</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(t)={\mathcal {L}}^{-1}\{H(s)\}={\frac {1}{\beta -\alpha }}\left(e^{-\alpha t}-e^{-\beta t}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2c81157d4756c96c422d5967320210df699ea8d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.667ex; height:5.676ex;" alt="{\displaystyle h(t)={\mathcal {L}}^{-1}\{H(s)\}={\frac {1}{\beta -\alpha }}\left(e^{-\alpha t}-e^{-\beta t}\right),}"> which is the impulse response of the system. <dl><dt>Convolution</dt></dl> The same result can be achieved using the <a href="/wiki/Convolution_theorem" title="Convolution theorem">convolution property</a> as if the system is a series of filters with transfer functions 1/(s + α) and 1/(s + β). That is, the inverse of <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(s)={\frac {1}{(s+\alpha )(s+\beta )}}={\frac {1}{s+\alpha }}\cdot {\frac {1}{s+\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>+</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>+</mo> <mi>β</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(s)={\frac {1}{(s+\alpha )(s+\beta )}}={\frac {1}{s+\alpha }}\cdot {\frac {1}{s+\beta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f83c812b257b87a39edd6cdd0c79826b61e9ad32" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.329ex; height:6.009ex;" alt="{\displaystyle H(s)={\frac {1}{(s+\alpha )(s+\beta )}}={\frac {1}{s+\alpha }}\cdot {\frac {1}{s+\beta }}}"> is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{-1}\!\left\{{\frac {1}{s+\alpha }}\right\}*{\mathcal {L}}^{-1}\!\left\{{\frac {1}{s+\beta }}\right\}=e^{-\alpha t}*e^{-\beta t}=\int _{0}^{t}e^{-\alpha x}e^{-\beta (t-x)}\,dx={\frac {e^{-\alpha t}-e^{-\beta t}}{\beta -\alpha }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="negativethinmathspace" /> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>+</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>}</mo> </mrow> <mo>∗</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="negativethinmathspace" /> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>+</mo> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>t</mi> </mrow> </msup> <mo>∗</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mi>t</mi> </mrow> </msup> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>x</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>t</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mi>t</mi> </mrow> </msup> </mrow> <mrow> <mi>β</mi> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{-1}\!\left\{{\frac {1}{s+\alpha }}\right\}*{\mathcal {L}}^{-1}\!\left\{{\frac {1}{s+\beta }}\right\}=e^{-\alpha t}*e^{-\beta t}=\int _{0}^{t}e^{-\alpha x}e^{-\beta (t-x)}\,dx={\frac {e^{-\alpha t}-e^{-\beta t}}{\beta -\alpha }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62edb0aca04bd02425ae546219c81ad7ab06b65a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:80.771ex; height:6.343ex;" alt="{\displaystyle {\mathcal {L}}^{-1}\!\left\{{\frac {1}{s+\alpha }}\right\}*{\mathcal {L}}^{-1}\!\left\{{\frac {1}{s+\beta }}\right\}=e^{-\alpha t}*e^{-\beta t}=\int _{0}^{t}e^{-\alpha x}e^{-\beta (t-x)}\,dx={\frac {e^{-\alpha t}-e^{-\beta t}}{\beta -\alpha }}.}"> <div class="mw-heading mw-heading3"><h3 id="Phase_delay">Phase delay</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=26" title="Edit section: Phase delay">edit</a>]</div> <table class="wikitable"> <tbody><tr> <th scope="col">Time function </th> <th scope="col">Laplace transform </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin {(\omega t+\varphi )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin {(\omega t+\varphi )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99fa02d82ae510003ae9e9045febe28fb4f9b403" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.698ex; height:2.843ex;" alt="{\displaystyle \sin {(\omega t+\varphi )}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {s\sin(\varphi )+\omega \cos(\varphi )}{s^{2}+\omega ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ω</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {s\sin(\varphi )+\omega \cos(\varphi )}{s^{2}+\omega ^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37bbb5e5bebb5142164448c3cc5adf5a1e794526" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.613ex; height:6.176ex;" alt="{\displaystyle {\frac {s\sin(\varphi )+\omega \cos(\varphi )}{s^{2}+\omega ^{2}}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {(\omega t+\varphi )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos {(\omega t+\varphi )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0babc64f2fdd9cb9c485567f05fa722ed847cf8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.953ex; height:2.843ex;" alt="{\displaystyle \cos {(\omega t+\varphi )}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {s\cos(\varphi )-\omega \sin(\varphi )}{s^{2}+\omega ^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ω</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {s\cos(\varphi )-\omega \sin(\varphi )}{s^{2}+\omega ^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46b359723cc4e040248e21aec40f4119d350aa96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.259ex; height:6.176ex;" alt="{\displaystyle {\frac {s\cos(\varphi )-\omega \sin(\varphi )}{s^{2}+\omega ^{2}}}.}"> </td></tr></tbody></table> Starting with the Laplace transform, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(s)={\frac {s\sin(\varphi )+\omega \cos(\varphi )}{s^{2}+\omega ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ω</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(s)={\frac {s\sin(\varphi )+\omega \cos(\varphi )}{s^{2}+\omega ^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f327d4b7ef4c3be5cd69e1916770a785d3357a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.591ex; height:6.176ex;" alt="{\displaystyle X(s)={\frac {s\sin(\varphi )+\omega \cos(\varphi )}{s^{2}+\omega ^{2}}}}"> we find the inverse by first rearranging terms in the fraction: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}X(s)&={\frac {s\sin(\varphi )}{s^{2}+\omega ^{2}}}+{\frac {\omega \cos(\varphi )}{s^{2}+\omega ^{2}}}\\&=\sin(\varphi )\left({\frac {s}{s^{2}+\omega ^{2}}}\right)+\cos(\varphi )\left({\frac {\omega }{s^{2}+\omega ^{2}}}\right).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>X</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ω</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}X(s)&={\frac {s\sin(\varphi )}{s^{2}+\omega ^{2}}}+{\frac {\omega \cos(\varphi )}{s^{2}+\omega ^{2}}}\\&=\sin(\varphi )\left({\frac {s}{s^{2}+\omega ^{2}}}\right)+\cos(\varphi )\left({\frac {\omega }{s^{2}+\omega ^{2}}}\right).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e964e4a6a345f724b029762f74452aefe43047bc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:49.489ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}X(s)&={\frac {s\sin(\varphi )}{s^{2}+\omega ^{2}}}+{\frac {\omega \cos(\varphi )}{s^{2}+\omega ^{2}}}\\&=\sin(\varphi )\left({\frac {s}{s^{2}+\omega ^{2}}}\right)+\cos(\varphi )\left({\frac {\omega }{s^{2}+\omega ^{2}}}\right).\end{aligned}}}"> We are now able to take the inverse Laplace transform of our terms: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x(t)&=\sin(\varphi ){\mathcal {L}}^{-1}\left\{{\frac {s}{s^{2}+\omega ^{2}}}\right\}+\cos(\varphi ){\mathcal {L}}^{-1}\left\{{\frac {\omega }{s^{2}+\omega ^{2}}}\right\}\\&=\sin(\varphi )\cos(\omega t)+\cos(\varphi )\sin(\omega t).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>}</mo> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x(t)&=\sin(\varphi ){\mathcal {L}}^{-1}\left\{{\frac {s}{s^{2}+\omega ^{2}}}\right\}+\cos(\varphi ){\mathcal {L}}^{-1}\left\{{\frac {\omega }{s^{2}+\omega ^{2}}}\right\}\\&=\sin(\varphi )\cos(\omega t)+\cos(\varphi )\sin(\omega t).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add01b6fdbc411017b6baaf9cabe6e7cad8a1e29" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:55.557ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}x(t)&=\sin(\varphi ){\mathcal {L}}^{-1}\left\{{\frac {s}{s^{2}+\omega ^{2}}}\right\}+\cos(\varphi ){\mathcal {L}}^{-1}\left\{{\frac {\omega }{s^{2}+\omega ^{2}}}\right\}\\&=\sin(\varphi )\cos(\omega t)+\cos(\varphi )\sin(\omega t).\end{aligned}}}"> This is just the <a href="/wiki/Trigonometric_identity#Angle_sum_and_difference_identities" class="mw-redirect" title="Trigonometric identity">sine of the sum</a> of the arguments, yielding: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)=\sin(\omega t+\varphi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)=\sin(\omega t+\varphi ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa38d219be9f0796685524de8b17b062220aa546" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.035ex; height:2.843ex;" alt="{\displaystyle x(t)=\sin(\omega t+\varphi ).}"> We can apply similar logic to find that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{-1}\left\{{\frac {s\cos \varphi -\omega \sin \varphi }{s^{2}+\omega ^{2}}}\right\}=\cos {(\omega t+\varphi )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>ω</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{-1}\left\{{\frac {s\cos \varphi -\omega \sin \varphi }{s^{2}+\omega ^{2}}}\right\}=\cos {(\omega t+\varphi )}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0af87b7716ff10ed13a077cdd65f9a8949bc910c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.277ex; height:6.176ex;" alt="{\displaystyle {\mathcal {L}}^{-1}\left\{{\frac {s\cos \varphi -\omega \sin \varphi }{s^{2}+\omega ^{2}}}\right\}=\cos {(\omega t+\varphi )}.}"> <div class="mw-heading mw-heading3"><h3 id="Statistical_mechanics">Statistical mechanics</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=27" title="Edit section: Statistical mechanics">edit</a>]</div> In <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, the Laplace transform of the density of states <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(E)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cc6f6fcf06a7c595416e9f92905fa4936751fa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.701ex; height:2.843ex;" alt="{\displaystyle g(E)}"> defines the <a href="/wiki/Partition_function_(statistical_mechanics)" title="Partition function (statistical mechanics)">partition function</a>.<a href="#cite_note-39">[39]</a> That is, the canonical partition function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7c552cea5b10b472898a6985fba0fb724941076" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle Z(\beta )}"> is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(\beta )=\int _{0}^{\infty }e^{-\beta E}g(E)\,dE}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mi>E</mi> </mrow> </msup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(\beta )=\int _{0}^{\infty }e^{-\beta E}g(E)\,dE}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64dddd428c3a83473b8851f2a587ac558806dd56" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.518ex; height:5.843ex;" alt="{\displaystyle Z(\beta )=\int _{0}^{\infty }e^{-\beta E}g(E)\,dE}"> and the inverse is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(E)={\frac {1}{2\pi i}}\int _{\beta _{0}-i\infty }^{\beta _{0}+i\infty }e^{\beta E}Z(\beta )\,d\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>i</mi> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>E</mi> </mrow> </msup> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(E)={\frac {1}{2\pi i}}\int _{\beta _{0}-i\infty }^{\beta _{0}+i\infty }e^{\beta E}Z(\beta )\,d\beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c45a4434f38900e5c0a57efc6e206ebba22922f9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:31.924ex; height:6.843ex;" alt="{\displaystyle g(E)={\frac {1}{2\pi i}}\int _{\beta _{0}-i\infty }^{\beta _{0}+i\infty }e^{\beta E}Z(\beta )\,d\beta }"> <div class="mw-heading mw-heading3"><h3 id="Spatial_(not_time)_structure_from_astronomical_spectrum">Spatial (not time) structure from astronomical spectrum</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=28" title="Edit section: Spatial (not time) structure from astronomical spectrum">edit</a>]</div> The wide and general applicability of the Laplace transform and its inverse is illustrated by an application in astronomy which provides some information on the spatial distribution of matter of an <a href="/wiki/Astronomy" title="Astronomy">astronomical</a> source of <a href="/wiki/Radiofrequency" class="mw-redirect" title="Radiofrequency">radiofrequency</a> <a href="/wiki/Thermal_radiation" title="Thermal radiation">thermal radiation</a> too distant to <a href="/wiki/Angular_resolution" title="Angular resolution">resolve</a> as more than a point, given its <a href="/wiki/Flux_density" class="mw-redirect" title="Flux density">flux density</a> <a href="/wiki/Spectrum" title="Spectrum">spectrum</a>, rather than relating the time domain with the spectrum (frequency domain). Assuming certain properties of the object, e.g. spherical shape and constant temperature, calculations based on carrying out an inverse Laplace transformation on the spectrum of the object can produce the only possible <a href="/wiki/Mathematical_model" title="Mathematical model">model</a> of the distribution of matter in it (density as a function of distance from the center) consistent with the spectrum.<a href="#cite_note-40">[40]</a> When independent information on the structure of an object is available, the inverse Laplace transform method has been found to be in good agreement. <div class="mw-heading mw-heading3"><h3 id="Birth_and_death_processes">Birth and death processes</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=29" title="Edit section: Birth and death processes">edit</a>]</div> Consider a <a href="/wiki/Random_walk" title="Random walk">random walk</a>, with steps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{+1,-1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{+1,-1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99c2868081b5f246c41752f501c5eebebe0f94b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.3ex; height:2.843ex;" alt="{\displaystyle \{+1,-1\}}"> occurring with probabilities <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,q=1-p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p,q=1-p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e1199fd05c8effa0070f33fcc7f781bd87420e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.633ex; height:2.509ex;" alt="{\displaystyle p,q=1-p}">.<a href="#cite_note-41">[41]</a> Suppose also that the time step is an <a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a>, with parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }">. Then the probability of the walk being at the lattice point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}(t)=\int _{0}^{t}\lambda e^{-\lambda (t-s)}(pP_{n-1}(s)+qP_{n+1}(s))\,ds\quad (+e^{-\lambda t}\quad {\text{when}}\ n=0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>λ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>q</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> <mspace width="1em" /> <mo stretchy="false">(</mo> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> <mi>t</mi> </mrow> </msup> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>when</mtext> </mrow> <mtext> </mtext> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}(t)=\int _{0}^{t}\lambda e^{-\lambda (t-s)}(pP_{n-1}(s)+qP_{n+1}(s))\,ds\quad (+e^{-\lambda t}\quad {\text{when}}\ n=0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a0959f6295bf88062fde4ba4b6ac0f81bcbc6a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:70.557ex; height:6.176ex;" alt="{\displaystyle P_{n}(t)=\int _{0}^{t}\lambda e^{-\lambda (t-s)}(pP_{n-1}(s)+qP_{n+1}(s))\,ds\quad (+e^{-\lambda t}\quad {\text{when}}\ n=0).}"></dd></dl> This leads to a system of <a href="/wiki/Integral_equation" title="Integral equation">integral equations</a> (or equivalently a system of differential equations). However, because it is a system of convolution equations, the Laplace transform converts it into a system of linear equations for <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{n}(s)={\mathcal {L}}(P_{n})(s),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{n}(s)={\mathcal {L}}(P_{n})(s),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a72a4be44204142fa791f7eb09cd2c1bee75320a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.212ex; height:2.843ex;" alt="{\displaystyle \pi _{n}(s)={\mathcal {L}}(P_{n})(s),}"></dd></dl> namely: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{n}(s)={\frac {\lambda }{\lambda +s}}(p\pi _{n-1}(s)+q\pi _{n+1}(s))\quad (+{\frac {1}{\lambda +s}}\quad {\text{when}}\ n=0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mrow> <mi>λ</mi> <mo>+</mo> <mi>s</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>q</mi> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>λ</mi> <mo>+</mo> <mi>s</mi> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>when</mtext> </mrow> <mtext> </mtext> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{n}(s)={\frac {\lambda }{\lambda +s}}(p\pi _{n-1}(s)+q\pi _{n+1}(s))\quad (+{\frac {1}{\lambda +s}}\quad {\text{when}}\ n=0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f907c5e590245abd2c8e0870a68b5b11bb84693d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:62.557ex; height:5.676ex;" alt="{\displaystyle \pi _{n}(s)={\frac {\lambda }{\lambda +s}}(p\pi _{n-1}(s)+q\pi _{n+1}(s))\quad (+{\frac {1}{\lambda +s}}\quad {\text{when}}\ n=0)}"></dd></dl> which may now be solved by standard methods. <div class="mw-heading mw-heading3"><h3 id="Tauberian_theory">Tauberian theory</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=30" title="Edit section: Tauberian theory">edit</a>]</div> The Laplace transform of the measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\infty )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc2d914c2df66bc0f7893bfb8da36766650fe47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle [0,\infty )}"> is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}\mu (s)=\int _{0}^{\infty }e^{-st}d\mu (t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\mu (s)=\int _{0}^{\infty }e^{-st}d\mu (t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9afbafabe4b90d6f7534beead22f30b7a6af667" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.603ex; height:5.843ex;" alt="{\displaystyle {\mathcal {L}}\mu (s)=\int _{0}^{\infty }e^{-st}d\mu (t).}"></dd></dl> It is intuitively clear that, for small <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76beea94b6662bd490c61c0628dddd8a8cd35538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s>0}">, the exponentially decaying integrand will become more sensitive to the concentration of the measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> on larger subsets of the domain. To make this more precise, introduce the distribution function: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(t)=\mu ([0,t)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(t)=\mu ([0,t)).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea188d9526432605026c10b62b60ea358f3cc99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.635ex; height:2.843ex;" alt="{\displaystyle M(t)=\mu ([0,t)).}"></dd></dl> Formally, we expect a limit of the following kind: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{s\to 0^{+}}{\mathcal {L}}\mu (s)=\lim _{t\to \infty }M(t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>M</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{s\to 0^{+}}{\mathcal {L}}\mu (s)=\lim _{t\to \infty }M(t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee1a67ea273361fb443854092f18c2dbc8e14312" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.834ex; height:4.343ex;" alt="{\displaystyle \lim _{s\to 0^{+}}{\mathcal {L}}\mu (s)=\lim _{t\to \infty }M(t).}"></dd></dl> <a href="/wiki/Tauberian_theorem" class="mw-redirect" title="Tauberian theorem">Tauberian theorems</a> are theorems relating the asymptotics of the Laplace transform, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\to 0^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\to 0^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9679f90f65dda0130f75c5cddf130f859cb51ed3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.378ex; height:2.509ex;" alt="{\displaystyle s\to 0^{+}}">, to those of the distribution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a34d7a61899d577d950881b4a44888d43f3fa93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.777ex; height:2.009ex;" alt="{\displaystyle t\to \infty }">. They are thus of importance in asymptotic formulae of <a href="/wiki/Probability" title="Probability">probability</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, where often the spectral side has asymptotics that are simpler to infer.<a href="#cite_note-42">[42]</a> Two tauberian theorems of note are the <a href="/wiki/Hardy%E2%80%93Littlewood_tauberian_theorem" class="mw-redirect" title="Hardy–Littlewood tauberian theorem">Hardy–Littlewood tauberian theorem</a> and the <a href="/wiki/Wiener_tauberian_theorem" class="mw-redirect" title="Wiener tauberian theorem">Wiener tauberian theorem</a>. The Wiener theorem generalizes the <a href="/wiki/Ikehara_tauberian_theorem" class="mw-redirect" title="Ikehara tauberian theorem">Ikehara tauberian theorem</a>, which is the following statement: Let A(x) be a non-negative, <a href="/wiki/Monotonic_function" title="Monotonic function">monotonic</a> nondecreasing function of x, defined for 0 ≤ x < ∞. Suppose that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(s)=\int _{0}^{\infty }A(x)e^{-xs}\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> <mi>s</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(s)=\int _{0}^{\infty }A(x)e^{-xs}\,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60205a12b7b14fd767cb697f4292a1d332ca03c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.124ex; height:5.843ex;" alt="{\displaystyle f(s)=\int _{0}^{\infty }A(x)e^{-xs}\,dx}"></dd></dl> converges for ℜ(s) > 1 to the function ƒ(s) and that, for some non-negative number c, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(s)-{\frac {c}{s-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(s)-{\frac {c}{s-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76ebfebe921c7680c0d3eeca5e46c83b77f1c7ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.948ex; height:4.843ex;" alt="{\displaystyle f(s)-{\frac {c}{s-1}}}"></dd></dl> has an extension as a <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a> for ℜ(s) ≥ 1. Then the <a href="/wiki/Limit_of_a_function" title="Limit of a function">limit</a> as x goes to infinity of e−x A(x) is equal to c. This statement can be applied in particular to the <a href="/wiki/Logarithmic_derivative" title="Logarithmic derivative">logarithmic derivative</a> of <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>, and thus provides an extremely short way to prove the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a>.<a href="#cite_note-43">[43]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=31" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output 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signal processing</a></li> <li><a href="/wiki/Bernstein%27s_theorem_on_monotone_functions" title="Bernstein's theorem on monotone functions">Bernstein's theorem on monotone functions</a></li> <li><a href="/wiki/Continuous-repayment_mortgage#Mortgage_difference_and_differential_equation" title="Continuous-repayment mortgage">Continuous-repayment mortgage</a></li> <li><a href="/wiki/Hamburger_moment_problem" title="Hamburger moment problem">Hamburger moment problem</a></li> <li><a href="/wiki/Hardy%E2%80%93Littlewood_Tauberian_theorem" title="Hardy–Littlewood Tauberian theorem">Hardy–Littlewood Tauberian theorem</a></li> <li><a href="/wiki/Laplace%E2%80%93Carson_transform" title="Laplace–Carson transform">Laplace–Carson transform</a></li> <li><a href="/wiki/Moment-generating_function" title="Moment-generating function">Moment-generating function</a></li> <li><a href="/wiki/Nonlocal_operator" title="Nonlocal operator">Nonlocal operator</a></li> <li><a href="/wiki/Post%27s_inversion_formula" class="mw-redirect" title="Post's inversion formula">Post's inversion formula</a></li> <li><a href="/wiki/Signal-flow_graph" title="Signal-flow graph">Signal-flow graph</a></li> <li><a href="/wiki/Transfer_function" title="Transfer function">Transfer function</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=32" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Lynn_1986_pp._225–272-1"><a href="#cite_ref-Lynn_1986_pp._225–272_1-0">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLynn1986" class="citation book cs1">Lynn, Paul A. (1986). "The Laplace Transform and the z-transform". Electronic Signals and Systems. London: Macmillan Education UK. pp. 225–272. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-349-18461-3_6">10.1007/978-1-349-18461-3_6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-333-39164-8" title="Special:BookSources/978-0-333-39164-8"><bdi>978-0-333-39164-8</bdi></a>. <q>Laplace Transform and the z-transform are closely related to the Fourier Transform. Laplace Transform is somewhat more general in scope than the Fourier Transform, and is widely used by engineers for describing continuous circuits and systems, including automatic control systems.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Laplace+Transform+and+the+z-transform&rft.btitle=Electronic+Signals+and+Systems&rft.place=London&rft.pages=225-272&rft.pub=Macmillan+Education+UK&rft.date=1986&rft_id=info%3Adoi%2F10.1007%2F978-1-349-18461-3_6&rft.isbn=978-0-333-39164-8&rft.aulast=Lynn&rft.aufirst=Paul+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx">"Differential Equations – Laplace Transforms"</a>. 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Retrieved 2020-08-08.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Laplace+Transform&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FLaplaceTransform.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2 cs1-prop-foreign-lang-source">"Des Fonctions génératrices" [On generating functions], <a rel="nofollow" class="external text" href="https://archive.org/details/thorieanalytiqu01laplgoog">Théorie analytique des Probabilités</a> [Analytical Probability Theory] (in French) (2nd ed.), Paris, 1814, chap.I sect.2-20</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Des+Fonctions+g%C3%A9n%C3%A9ratrices&rft.btitle=Th%C3%A9orie+analytique+des+Probabilit%C3%A9s&rft.place=Paris&rft.pages=chap.I+sect.2-20&rft.edition=2nd&rft.date=1814&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fthorieanalytiqu01laplgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJaynes,_E._T._(Edwin_T.)2003" class="citation book cs1">Jaynes, E. T. (Edwin T.) (2003). Probability theory : the logic of science. Bretthorst, G. Larry. Cambridge, UK: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0511065892" title="Special:BookSources/0511065892"><bdi>0511065892</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/57254076">57254076</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+theory+%3A+the+logic+of+science&rft.place=Cambridge%2C+UK&rft.pub=Cambridge+University+Press&rft.date=2003&rft_id=info%3Aoclcnum%2F57254076&rft.isbn=0511065892&rft.au=Jaynes%2C+E.+T.+%28Edwin+T.%29&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbel1820" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Niels_Henrik_Abel" title="Niels Henrik Abel">Abel, Niels H.</a> (1820), "Sur les fonctions génératrices et leurs déterminantes", Œuvres Complètes (in French), vol. II (published 1839), pp. 77–88</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Sur+les+fonctions+g%C3%A9n%C3%A9ratrices+et+leurs+d%C3%A9terminantes&rft.btitle=%C5%92uvres+Compl%C3%A8tes&rft.pages=77-88&rft.date=1820&rft.aulast=Abel&rft.aufirst=Niels+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6FtDAQAAMAAJ&pg=RA2-PA67">1881 edition</a> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFEuler1744">Euler 1744</a>, <a href="#CITEREFEuler1753">Euler 1753</a>, <a href="#CITEREFEuler1769">Euler 1769</a> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFLagrange1773">Lagrange 1773</a> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#CITEREFGrattan-Guinness1997">Grattan-Guinness 1997</a>, p. 260 </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="#CITEREFGrattan-Guinness1997">Grattan-Guinness 1997</a>, p. 261 </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <a href="#CITEREFGrattan-Guinness1997">Grattan-Guinness 1997</a>, pp. 261–262 </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <a href="#CITEREFGrattan-Guinness1997">Grattan-Guinness 1997</a>, pp. 262–266 </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeaviside2008" class="citation cs2"><a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Heaviside, Oliver</a> (January 2008), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=y9auR0L6ZRcC&pg=PA234">"The solution of definite integrals by differential transformation"</a>, Electromagnetic Theory, vol. III, London, section 526, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781605206189" title="Special:BookSources/9781605206189"><bdi>9781605206189</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+solution+of+definite+integrals+by+differential+transformation&rft.btitle=Electromagnetic+Theory&rft.place=London&rft.pages=section+526&rft.date=2008-01&rft.isbn=9781605206189&rft.aulast=Heaviside&rft.aufirst=Oliver&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dy9auR0L6ZRcC%26pg%3DPA234&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>) </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGardnerBarnes1942" class="citation cs2">Gardner, Murray F.; Barnes, John L. (1942), Transients in Linear Systems studied by the Laplace Transform, New York: Wiley</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Transients+in+Linear+Systems+studied+by+the+Laplace+Transform&rft.place=New+York&rft.pub=Wiley&rft.date=1942&rft.aulast=Gardner&rft.aufirst=Murray+F.&rft.au=Barnes%2C+John+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988">, Appendix C </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLerch1903" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Mathias_Lerch" title="Mathias Lerch">Lerch, Mathias</a> (1903), "Sur un point de la théorie des fonctions génératrices d'Abel" [Proof of the inversion formula], <a href="/wiki/Acta_Mathematica" title="Acta Mathematica">Acta Mathematica</a> (in French), 27: 339–351, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02421315">10.1007/BF02421315</a>, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F501554">10338.dmlcz/501554</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Mathematica&rft.atitle=Sur+un+point+de+la+th%C3%A9orie+des+fonctions+g%C3%A9n%C3%A9ratrices+d%27Abel&rft.volume=27&rft.pages=339-351&rft.date=1903&rft_id=info%3Ahdl%2F10338.dmlcz%2F501554&rft_id=info%3Adoi%2F10.1007%2FBF02421315&rft.aulast=Lerch&rft.aufirst=Mathias&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBromwich1916" class="citation cs2"><a href="/wiki/Thomas_John_I%27Anson_Bromwich" title="Thomas John I'Anson Bromwich">Bromwich, Thomas J.</a> (1916), <a rel="nofollow" class="external text" href="https://zenodo.org/record/2319588">"Normal coordinates in dynamical systems"</a>, <a href="/wiki/Proceedings_of_the_London_Mathematical_Society" class="mw-redirect" title="Proceedings of the London Mathematical Society">Proceedings of the London Mathematical Society</a>, 15: 401–448, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs2-15.1.401">10.1112/plms/s2-15.1.401</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+London+Mathematical+Society&rft.atitle=Normal+coordinates+in+dynamical+systems&rft.volume=15&rft.pages=401-448&rft.date=1916&rft_id=info%3Adoi%2F10.1112%2Fplms%2Fs2-15.1.401&rft.aulast=Bromwich&rft.aufirst=Thomas+J.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F2319588&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> An influential book was: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGardnerBarnes1942" class="citation cs2">Gardner, Murray F.; Barnes, John L. (1942), Transients in Linear Systems studied by the Laplace Transform, New York: Wiley</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Transients+in+Linear+Systems+studied+by+the+Laplace+Transform&rft.place=New+York&rft.pub=Wiley&rft.date=1942&rft.aulast=Gardner&rft.aufirst=Murray+F.&rft.au=Barnes%2C+John+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDoetsch1937" class="citation cs2 cs1-prop-foreign-lang-source">Doetsch, Gustav (1937), Theorie und Anwendung der Laplacesche Transformation [Theory and Application of the Laplace Transform] (in German), Berlin: Springer</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theorie+und+Anwendung+der+Laplacesche+Transformation&rft.place=Berlin&rft.pub=Springer&rft.date=1937&rft.aulast=Doetsch&rft.aufirst=Gustav&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> translation 1943 </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <a href="#CITEREFFeller1971">Feller 1971</a>, §XIII.1. </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> The cumulative distribution function is the integral of the probability density function. </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMikusiński2014" class="citation book cs1">Mikusiński, Jan (14 July 2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=e8LSBQAAQBAJ">Operational Calculus</a>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781483278933" title="Special:BookSources/9781483278933"><bdi>9781483278933</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Operational+Calculus&rft.pub=Elsevier&rft.date=2014-07-14&rft.isbn=9781483278933&rft.aulast=Mikusi%C5%84ski&rft.aufirst=Jan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3De8LSBQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <a href="#CITEREFWidder1941">Widder 1941</a>, Chapter II, §1 </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <a href="#CITEREFWidder1941">Widder 1941</a>, Chapter VI, §2 </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <a href="#CITEREFKornKorn1967">Korn & Korn 1967</a>, pp. 226–227 </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <a href="#CITEREFBracewell2000">Bracewell 2000</a>, Table 14.1, p. 385 </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> Archived at <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211211/zvbdoSeGAgI">Ghostarchive</a> and the <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141220033002/https://www.youtube.com/watch?v=zvbdoSeGAgI&gl=US&hl=en">Wayback Machine</a>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMattuck" class="citation web cs1">Mattuck, Arthur. <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=zvbdoSeGAgI">"Where the Laplace Transform comes from"</a>. <a href="/wiki/YouTube" title="YouTube">YouTube</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=YouTube&rft.atitle=Where+the+Laplace+Transform+comes+from&rft.aulast=Mattuck&rft.aufirst=Arthur&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DzvbdoSeGAgI&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <a href="#CITEREFFeller1971">Feller 1971</a>, p. 432 </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaurent_Schwartz1966" class="citation book cs1"><a href="/wiki/Laurent_Schwartz" title="Laurent Schwartz">Laurent Schwartz</a> (1966). Mathematics for the physical sciences. Addison-Wesley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+for+the+physical+sciences&rft.pub=Addison-Wesley&rft.date=1966&rft.au=Laurent+Schwartz&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988">, p 224. </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTitchmarsh1986" class="citation cs2"><a href="/wiki/Edward_Charles_Titchmarsh" title="Edward Charles Titchmarsh">Titchmarsh, E.</a> (1986) [1948], Introduction to the theory of Fourier integrals (2nd ed.), <a href="/wiki/Clarendon_Press" class="mw-redirect" title="Clarendon Press">Clarendon Press</a>, p. 6, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8284-0324-5" title="Special:BookSources/978-0-8284-0324-5"><bdi>978-0-8284-0324-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+theory+of+Fourier+integrals&rft.pages=6&rft.edition=2nd&rft.pub=Clarendon+Press&rft.date=1986&rft.isbn=978-0-8284-0324-5&rft.aulast=Titchmarsh&rft.aufirst=E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <a href="#CITEREFTakacs1953">Takacs 1953</a>, p. 93 </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRileyHobsonBence2010" class="citation cs2">Riley, K. F.; Hobson, M. P.; Bence, S. J. (2010), Mathematical methods for physics and engineering (3rd ed.), Cambridge University Press, p. 455, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-86153-3" title="Special:BookSources/978-0-521-86153-3"><bdi>978-0-521-86153-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+methods+for+physics+and+engineering&rft.pages=455&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=2010&rft.isbn=978-0-521-86153-3&rft.aulast=Riley&rft.aufirst=K.+F.&rft.au=Hobson%2C+M.+P.&rft.au=Bence%2C+S.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDistefanoStubberudWilliams1995" class="citation cs2">Distefano, J. J.; Stubberud, A. R.; Williams, I. J. (1995), Feedback systems and control, Schaum's outlines (2nd ed.), McGraw-Hill, p. 78, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-017052-0" title="Special:BookSources/978-0-07-017052-0"><bdi>978-0-07-017052-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Feedback+systems+and+control&rft.series=Schaum%27s+outlines&rft.pages=78&rft.edition=2nd&rft.pub=McGraw-Hill&rft.date=1995&rft.isbn=978-0-07-017052-0&rft.aulast=Distefano&rft.aufirst=J.+J.&rft.au=Stubberud%2C+A.+R.&rft.au=Williams%2C+I.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLipschutzSpiegelLiu2009" class="citation book cs1">Lipschutz, S.; Spiegel, M. R.; Liu, J. (2009). Mathematical Handbook of Formulas and Tables. Schaum's Outline Series (3rd ed.). McGraw-Hill. p. 183. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-154855-7" title="Special:BookSources/978-0-07-154855-7"><bdi>978-0-07-154855-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Handbook+of+Formulas+and+Tables&rft.series=Schaum%27s+Outline+Series&rft.pages=183&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=2009&rft.isbn=978-0-07-154855-7&rft.aulast=Lipschutz&rft.aufirst=S.&rft.au=Spiegel%2C+M.+R.&rft.au=Liu%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> – provides the case for real q. </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <a rel="nofollow" class="external free" href="http://mathworld.wolfram.com/LaplaceTransform.html">http://mathworld.wolfram.com/LaplaceTransform.html</a> – Wolfram Mathword provides case for complex q </li> <li id="cite_note-FOOTNOTEBracewell1978227-35">^ <a href="#cite_ref-FOOTNOTEBracewell1978227_35-0">a</a> <a href="#cite_ref-FOOTNOTEBracewell1978227_35-1">b</a> <a href="#cite_ref-FOOTNOTEBracewell1978227_35-2">c</a> <a href="#cite_ref-FOOTNOTEBracewell1978227_35-3">d</a> <a href="#CITEREFBracewell1978">Bracewell 1978</a>, p. 227. </li> <li id="cite_note-FOOTNOTEWilliams197388-36">^ <a href="#cite_ref-FOOTNOTEWilliams197388_36-0">a</a> <a href="#cite_ref-FOOTNOTEWilliams197388_36-1">b</a> <a href="#cite_ref-FOOTNOTEWilliams197388_36-2">c</a> <a href="#CITEREFWilliams1973">Williams 1973</a>, p. 88. </li> <li id="cite_note-FOOTNOTEWilliams197389-37">^ <a href="#cite_ref-FOOTNOTEWilliams197389_37-0">a</a> <a href="#cite_ref-FOOTNOTEWilliams197389_37-1">b</a> <a href="#CITEREFWilliams1973">Williams 1973</a>, p. 89. </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <a href="#CITEREFKornKorn1967">Korn & Korn 1967</a>, §8.1 </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRK_PathriaPaul_Beal1996" class="citation book cs1">RK Pathria; Paul Beal (1996). <a rel="nofollow" class="external text" href="https://archive.org/details/statisticalmecha00path_911">Statistical mechanics</a> (2nd ed.). Butterworth-Heinemann. p. <a rel="nofollow" class="external text" href="https://archive.org/details/statisticalmecha00path_911/page/n66">56</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780750624695" title="Special:BookSources/9780750624695"><bdi>9780750624695</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+mechanics&rft.pages=56&rft.edition=2nd&rft.pub=Butterworth-Heinemann&rft.date=1996&rft.isbn=9780750624695&rft.au=RK+Pathria&rft.au=Paul+Beal&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fstatisticalmecha00path_911&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSalemSeaton1974" class="citation cs2">Salem, M.; Seaton, M. J. (1974), "I. Continuum spectra and brightness contours", <a href="/wiki/Monthly_Notices_of_the_Royal_Astronomical_Society" title="Monthly Notices of the Royal Astronomical Society">Monthly Notices of the Royal Astronomical Society</a>, 167: 493–510, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1974MNRAS.167..493S">1974MNRAS.167..493S</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fmnras%2F167.3.493">10.1093/mnras/167.3.493</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Monthly+Notices+of+the+Royal+Astronomical+Society&rft.atitle=I.+Continuum+spectra+and+brightness+contours&rft.volume=167&rft.pages=493-510&rft.date=1974&rft_id=info%3Adoi%2F10.1093%2Fmnras%2F167.3.493&rft_id=info%3Abibcode%2F1974MNRAS.167..493S&rft.aulast=Salem&rft.aufirst=M.&rft.au=Seaton%2C+M.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988">, and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSalem1974" class="citation cs2">Salem, M. (1974), "II. Three-dimensional models", Monthly Notices of the Royal Astronomical Society, 167: 511–516, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1974MNRAS.167..511S">1974MNRAS.167..511S</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fmnras%2F167.3.511">10.1093/mnras/167.3.511</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Monthly+Notices+of+the+Royal+Astronomical+Society&rft.atitle=II.+Three-dimensional+models&rft.volume=167&rft.pages=511-516&rft.date=1974&rft_id=info%3Adoi%2F10.1093%2Fmnras%2F167.3.511&rft_id=info%3Abibcode%2F1974MNRAS.167..511S&rft.aulast=Salem&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeller" class="citation book cs1">Feller. Introduction to Probability Theory, volume II,pp=479-483.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Probability+Theory%2C+volume+II%2Cpp%3D479-483&rft.au=Feller&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeller" class="citation book cs1">Feller. Introduction to Probability Theory, volume II,pp=479-483.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Probability+Theory%2C+volume+II%2Cpp%3D479-483&rft.au=Feller&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFS._Ikehara1931" class="citation cs2"><a href="/wiki/Shikao_Ikehara" title="Shikao Ikehara">S. Ikehara</a> (1931), "An extension of Landau's theorem in the analytic theory of numbers", Journal of Mathematics and Physics of the Massachusetts Institute of Technology, 10 (1–4): 1–12, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fsapm19311011">10.1002/sapm19311011</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0001.12902">0001.12902</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematics+and+Physics+of+the+Massachusetts+Institute+of+Technology&rft.atitle=An+extension+of+Landau%27s+theorem+in+the+analytic+theory+of+numbers&rft.volume=10&rft.issue=1%E2%80%934&rft.pages=1-12&rft.date=1931&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0001.12902%23id-name%3DZbl&rft_id=info%3Adoi%2F10.1002%2Fsapm19311011&rft.au=S.+Ikehara&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=33" title="Edit section: References">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Modern">Modern</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=34" title="Edit section: Modern">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBracewell1978" class="citation cs2">Bracewell, Ronald N. (1978), The Fourier Transform and its Applications (2nd ed.), McGraw-Hill Kogakusha, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-007013-4" title="Special:BookSources/978-0-07-007013-4"><bdi>978-0-07-007013-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Fourier+Transform+and+its+Applications&rft.edition=2nd&rft.pub=McGraw-Hill+Kogakusha&rft.date=1978&rft.isbn=978-0-07-007013-4&rft.aulast=Bracewell&rft.aufirst=Ronald+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBracewell2000" class="citation cs2">Bracewell, R. N. (2000), The Fourier Transform and Its Applications (3rd ed.), Boston: McGraw-Hill, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-116043-8" title="Special:BookSources/978-0-07-116043-8"><bdi>978-0-07-116043-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Fourier+Transform+and+Its+Applications&rft.place=Boston&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=2000&rft.isbn=978-0-07-116043-8&rft.aulast=Bracewell&rft.aufirst=R.+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeller1971" class="citation cs2"><a href="/wiki/William_Feller" title="William Feller">Feller, William</a> (1971), An introduction to probability theory and its applications. Vol. II., Second edition, New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0270403">0270403</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+probability+theory+and+its+applications.+Vol.+II.&rft.place=New+York&rft.series=Second+edition&rft.pub=John+Wiley+%26+Sons&rft.date=1971&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0270403%23id-name%3DMR&rft.aulast=Feller&rft.aufirst=William&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKornKorn1967" class="citation cs2">Korn, G. A.; <a href="/wiki/Theresa_M._Korn" title="Theresa M. Korn">Korn, T. M.</a> (1967), Mathematical Handbook for Scientists and Engineers (2nd ed.), McGraw-Hill Companies, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-035370-1" title="Special:BookSources/978-0-07-035370-1"><bdi>978-0-07-035370-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Handbook+for+Scientists+and+Engineers&rft.edition=2nd&rft.pub=McGraw-Hill+Companies&rft.date=1967&rft.isbn=978-0-07-035370-1&rft.aulast=Korn&rft.aufirst=G.+A.&rft.au=Korn%2C+T.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWidder1941" class="citation cs2">Widder, David Vernon (1941), The Laplace Transform, Princeton Mathematical Series, v. 6, <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0005923">0005923</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Laplace+Transform&rft.series=Princeton+Mathematical+Series%2C+v.+6&rft.pub=Princeton+University+Press&rft.date=1941&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0005923%23id-name%3DMR&rft.aulast=Widder&rft.aufirst=David+Vernon&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliams1973" class="citation cs2">Williams, J. (1973), Laplace Transforms, Problem Solvers, George Allen & Unwin, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-04-512021-5" title="Special:BookSources/978-0-04-512021-5"><bdi>978-0-04-512021-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Laplace+Transforms&rft.series=Problem+Solvers&rft.pub=George+Allen+%26+Unwin&rft.date=1973&rft.isbn=978-0-04-512021-5&rft.aulast=Williams&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTakacs1953" class="citation cs2 cs1-prop-foreign-lang-source">Takacs, J. (1953), "Fourier amplitudok meghatarozasa operatorszamitassal", Magyar Hiradastechnika (in Hungarian), IV (7–8): 93–96</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Magyar+Hiradastechnika&rft.atitle=Fourier+amplitudok+meghatarozasa+operatorszamitassal&rft.volume=IV&rft.issue=7%E2%80%938&rft.pages=93-96&rft.date=1953&rft.aulast=Takacs&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Historical">Historical</h3>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=35" title="Edit section: Historical">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEuler1744" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler, L.</a> (1744), "De constructione aequationum" [The Construction of Equations], Opera Omnia, 1st series (in Latin), 22: 150–161</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Opera+Omnia&rft.atitle=De+constructione+aequationum&rft.volume=22&rft.pages=150-161&rft.date=1744&rft.aulast=Euler&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEuler1753" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler, L.</a> (1753), "Methodus aequationes differentiales" [A Method for Solving Differential Equations], Opera Omnia, 1st series (in Latin), 22: 181–213</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Opera+Omnia&rft.atitle=Methodus+aequationes+differentiales&rft.volume=22&rft.pages=181-213&rft.date=1753&rft.aulast=Euler&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEuler1992" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler, L.</a> (1992) [1769], "Institutiones calculi integralis, Volume 2" [Institutions of Integral Calculus], Opera Omnia, 1st series (in Latin), 12, Basel: Birkhäuser, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3764314743" title="Special:BookSources/978-3764314743"><bdi>978-3764314743</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Opera+Omnia&rft.atitle=Institutiones+calculi+integralis%2C+Volume+2&rft.volume=12&rft.date=1992&rft.isbn=978-3764314743&rft.aulast=Euler&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988">, Chapters 3–5</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEuler1769" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler, Leonhard</a> (1769), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BFqWNwpfqo8C">Institutiones calculi integralis</a> [Institutions of Integral Calculus] (in Latin), vol. II, Paris: Petropoli, ch. 3–5, pp. 57–153</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Institutiones+calculi+integralis&rft.place=Paris&rft.pages=ch.+3-5%2C+pp.+57-153&rft.pub=Petropoli&rft.date=1769&rft.aulast=Euler&rft.aufirst=Leonhard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBFqWNwpfqo8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrattan-Guinness1997" class="citation cs2"><a href="/wiki/Ivor_Grattan-Guinness" title="Ivor Grattan-Guinness">Grattan-Guinness, I</a> (1997), "Laplace's integral solutions to partial differential equations", in Gillispie, C. C. (ed.), Pierre Simon Laplace 1749–1827: A Life in Exact Science, Princeton: Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-01185-1" title="Special:BookSources/978-0-691-01185-1"><bdi>978-0-691-01185-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Laplace%27s+integral+solutions+to+partial+differential+equations&rft.btitle=Pierre+Simon+Laplace+1749%E2%80%931827%3A+A+Life+in+Exact+Science&rft.place=Princeton&rft.pub=Princeton+University+Press&rft.date=1997&rft.isbn=978-0-691-01185-1&rft.aulast=Grattan-Guinness&rft.aufirst=I&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLagrange1773" class="citation cs2"><a href="/wiki/Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Lagrange, J. L.</a> (1773), Mémoire sur l'utilité de la méthode, Œuvres de Lagrange, vol. 2, pp. 171–234</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=M%C3%A9moire+sur+l%27utilit%C3%A9+de+la+m%C3%A9thode&rft.series=%C5%92uvres+de+Lagrange&rft.pages=171-234&rft.date=1773&rft.aulast=Lagrange&rft.aufirst=J.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Laplace_transform&action=edit&section=36" title="Edit section: Further reading">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Arendt, Wolfgang; Batty, Charles J.K.; Hieber, Matthias; Neubrander, Frank (2002), Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser Basel, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-6549-3" title="Special:BookSources/978-3-7643-6549-3"><bdi>978-3-7643-6549-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vector-Valued+Laplace+Transforms+and+Cauchy+Problems&rft.pub=Birkh%C3%A4user+Basel&rft.date=2002&rft.isbn=978-3-7643-6549-3&rft.aulast=Arendt&rft.aufirst=Wolfgang&rft.au=Batty%2C+Charles+J.K.&rft.au=Hieber%2C+Matthias&rft.au=Neubrander%2C+Frank&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Davies, Brian (2002), Integral transforms and their applications (Third ed.), New York: Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95314-4" title="Special:BookSources/978-0-387-95314-4"><bdi>978-0-387-95314-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Integral+transforms+and+their+applications&rft.place=New+York&rft.edition=Third&rft.pub=Springer&rft.date=2002&rft.isbn=978-0-387-95314-4&rft.aulast=Davies&rft.aufirst=Brian&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Deakin, M. A. B. (1981), "The development of the Laplace transform", Archive for History of Exact Sciences, 25 (4): 343–390, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01395660">10.1007/BF01395660</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:117913073">117913073</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=The+development+of+the+Laplace+transform&rft.volume=25&rft.issue=4&rft.pages=343-390&rft.date=1981&rft_id=info%3Adoi%2F10.1007%2FBF01395660&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A117913073%23id-name%3DS2CID&rft.aulast=Deakin&rft.aufirst=M.+A.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Deakin, M. A. B. (1982), "The development of the Laplace transform", Archive for History of Exact Sciences, 26 (4): 351–381, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00418754">10.1007/BF00418754</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123071842">123071842</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=The+development+of+the+Laplace+transform&rft.volume=26&rft.issue=4&rft.pages=351-381&rft.date=1982&rft_id=info%3Adoi%2F10.1007%2FBF00418754&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123071842%23id-name%3DS2CID&rft.aulast=Deakin&rft.aufirst=M.+A.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/Gustav_Doetsch" title="Gustav Doetsch">Doetsch, Gustav</a> (1974), Introduction to the Theory and Application of the Laplace Transformation, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-06407-9" title="Special:BookSources/978-0-387-06407-9"><bdi>978-0-387-06407-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+Theory+and+Application+of+the+Laplace+Transformation&rft.pub=Springer&rft.date=1974&rft.isbn=978-0-387-06407-9&rft.aulast=Doetsch&rft.aufirst=Gustav&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li>Mathews, Jon; Walker, Robert L. (1970), Mathematical methods of physics (2nd ed.), New York: W. A. Benjamin, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8053-7002-1" title="Special:BookSources/0-8053-7002-1">0-8053-7002-1</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Polyanin, A. D.; Manzhirov, A. V. (1998), Handbook of Integral Equations, Boca Raton: CRC Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8493-2876-3" title="Special:BookSources/978-0-8493-2876-3"><bdi>978-0-8493-2876-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Integral+Equations&rft.place=Boca+Raton&rft.pub=CRC+Press&rft.date=1998&rft.isbn=978-0-8493-2876-3&rft.aulast=Polyanin&rft.aufirst=A.+D.&rft.au=Manzhirov%2C+A.+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Laurent_Schwartz" title="Laurent Schwartz">Schwartz, Laurent</a> (1952), "Transformation de Laplace des distributions", Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] (in French), 1952: 196–206, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0052555">0052555</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comm.+S%C3%A9m.+Math.+Univ.+Lund+%5BMedd.+Lunds+Univ.+Mat.+Sem.%5D&rft.atitle=Transformation+de+Laplace+des+distributions&rft.volume=1952&rft.pages=196-206&rft.date=1952&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0052555%23id-name%3DMR&rft.aulast=Schwartz&rft.aufirst=Laurent&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/Laurent_Schwartz" title="Laurent Schwartz">Schwartz, Laurent</a> (2008) [1966], <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-_AuDQAAQBAJ&pg=PA215">Mathematics for the Physical Sciences</a>, Dover Books on Mathematics, New York: Dover Publications, pp. 215–241, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-46662-0" title="Special:BookSources/978-0-486-46662-0"><bdi>978-0-486-46662-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+for+the+Physical+Sciences&rft.place=New+York&rft.series=Dover+Books+on+Mathematics&rft.pages=215-241&rft.pub=Dover+Publications&rft.date=2008&rft.isbn=978-0-486-46662-0&rft.aulast=Schwartz&rft.aufirst=Laurent&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-_AuDQAAQBAJ%26pg%3DPA215&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"> - See Chapter VI. The Laplace transform.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Siebert, William McC. (1986), Circuits, Signals, and Systems, Cambridge, Massachusetts: MIT Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-19229-3" title="Special:BookSources/978-0-262-19229-3"><bdi>978-0-262-19229-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Circuits%2C+Signals%2C+and+Systems&rft.place=Cambridge%2C+Massachusetts&rft.pub=MIT+Press&rft.date=1986&rft.isbn=978-0-262-19229-3&rft.aulast=Siebert&rft.aufirst=William+McC.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Widder, David Vernon (1945), "What is the Laplace transform?", <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">The American Mathematical Monthly</a>, 52 (8): 419–425, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2305640">10.2307/2305640</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9890">0002-9890</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2305640">2305640</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0013447">0013447</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=What+is+the+Laplace+transform%3F&rft.volume=52&rft.issue=8&rft.pages=419-425&rft.date=1945&rft.issn=0002-9890&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0013447%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2305640%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2305640&rft.aulast=Widder&rft.aufirst=David+Vernon&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li>J.A.C.Weidman and Bengt Fornberg: "Fully numerical Laplace transform methods", Numerical Algorithms, vol.92 (2023), pp. 985–1006. <a rel="nofollow" class="external free" href="https://doi.org/10.1007/s11075-022-01368-x">https://doi.org/10.1007/s11075-022-01368-x</a> .</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a 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of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Laplace+transform&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DLaplace_transform&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://wims.unice.fr/wims/wims.cgi?lang=en&+module=tool%2Fanalysis%2Ffourierlaplace">Online Computation</a> of the transform or inverse transform, wims.unice.fr</li> <li><a rel="nofollow" class="external text" href="http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm">Tables of Integral Transforms</a> at EqWorld: The World of Mathematical Equations.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/LaplaceTransform.html">"Laplace Transform"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Laplace+Transform&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FLaplaceTransform.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaplace+transform" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/">Good explanations of the initial and final value theorems</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090108132440/http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/">Archived</a> 2009-01-08 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://www.mathpages.com/home/kmath508/kmath508.htm">Laplace Transforms</a> at MathPages</li> <li><a rel="nofollow" class="external text" href="http://www.wolframalpha.com/input/?i=laplace+transform+example">Computational Knowledge Engine</a> allows to easily calculate Laplace Transforms and its inverse Transform.</li> <li><a rel="nofollow" class="external text" href="http://www.laplacetransformcalculator.com/easy-laplace-transform-calculator/">Laplace Calculator</a> to calculate Laplace Transforms online easily.</li> <li><a rel="nofollow" class="external text" href="https://johnflux.com/2019/02/12/laplace-transform-visualized/">Code to visualize Laplace Transforms</a> and many example videos.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist 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Laplace transform - Wikipedia