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mw-first-heading"><span class="mw-page-title-main">Zeros and poles</span></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. 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interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D1%8E%D1%81_(%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%B5%D0%BD_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7)" title="Полюс (комплексен анализ) – Bulgarian" lang="bg" hreflang="bg" data-title="Полюс (комплексен анализ)" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Pol_(an%C3%A0lisi_complexa)" title="Pol (anàlisi complexa) – Catalan" lang="ca" hreflang="ca" data-title="Pol (anàlisi complexa)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Pol_(matematisk)" title="Pol (matematisk) – Danish" lang="da" hreflang="da" data-title="Pol (matematisk)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Polstelle" title="Polstelle – German" lang="de" hreflang="de" data-title="Polstelle" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Poolus_(kompleksmuutuja_funktsiooniteooria)" title="Poolus (kompleksmuutuja funktsiooniteooria) – Estonian" lang="et" hreflang="et" data-title="Poolus (kompleksmuutuja funktsiooniteooria)" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CF%8C%CE%BB%CE%BF%CF%82_(%CE%BC%CE%B9%CE%B3%CE%B1%CE%B4%CE%B9%CE%BA%CE%AE_%CE%B1%CE%BD%CE%AC%CE%BB%CF%85%CF%83%CE%B7)" title="Πόλος (μιγαδική ανάλυση) – Greek" lang="el" hreflang="el" data-title="Πόλος (μιγαδική ανάλυση)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Polo_(an%C3%A1lisis_complejo)" title="Polo (análisis complejo) – Spanish" lang="es" hreflang="es" data-title="Polo (análisis complejo)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Poluso_(kompleksa_analitiko)" title="Poluso (kompleksa analitiko) – Esperanto" lang="eo" hreflang="eo" data-title="Poluso (kompleksa analitiko)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B7%D8%A8_(%D8%A2%D9%86%D8%A7%D9%84%DB%8C%D8%B2_%D9%85%D8%AE%D8%AA%D9%84%D8%B7)" title="قطب (آنالیز مختلط) – Persian" lang="fa" hreflang="fa" data-title="قطب (آنالیز مختلط)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/P%C3%B4le_(math%C3%A9matiques)" title="Pôle (mathématiques) – French" lang="fr" hreflang="fr" data-title="Pôle (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B7%B9%EC%A0%90_(%EB%B3%B5%EC%86%8C%ED%95%B4%EC%84%9D%ED%95%99)" title="극점 (복소해석학) – Korean" lang="ko" hreflang="ko" data-title="극점 (복소해석학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Polo_(analisi_complessa)" title="Polo (analisi complessa) – Italian" lang="it" hreflang="it" data-title="Polo (analisi complessa)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%95%D7%98%D7%91_(%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%A8%D7%95%D7%9B%D7%91%D7%AA)" title="קוטב (אנליזה מרוכבת) – Hebrew" lang="he" hreflang="he" data-title="קוטב (אנליזה מרוכבת)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Polius_(kompleksin%C4%97_analiz%C4%97)" title="Polius (kompleksinė analizė) – Lithuanian" lang="lt" hreflang="lt" data-title="Polius (kompleksinė analizė)" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/P%C3%B3lus_(komplex_anal%C3%ADzis)" title="Pólus (komplex analízis) – Hungarian" lang="hu" hreflang="hu" data-title="Pólus (komplex analízis)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Pool_(functietheorie)" title="Pool (functietheorie) – Dutch" lang="nl" hreflang="nl" data-title="Pool (functietheorie)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%A5%B5_(%E8%A4%87%E7%B4%A0%E8%A7%A3%E6%9E%90)" title="極 (複素解析) – Japanese" lang="ja" hreflang="ja" data-title="極 (複素解析)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Pol_i_kompleks_analyse" title="Pol i kompleks analyse – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Pol i kompleks analyse" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Biegun_(analiza_zespolona)" title="Biegun (analiza zespolona) – Polish" lang="pl" hreflang="pl" data-title="Biegun (analiza zespolona)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Polo_(an%C3%A1lise_complexa)" title="Polo (análise complexa) – Portuguese" lang="pt" hreflang="pt" data-title="Polo (análise complexa)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Pol_(matematic%C4%83)" title="Pol (matematică) – Romanian" lang="ro" hreflang="ro" data-title="Pol (matematică)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D1%8E%D1%81_(%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D1%8B%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7)" title="Полюс (комплексный анализ) – Russian" lang="ru" hreflang="ru" data-title="Полюс (комплексный анализ)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Pol_(kompleksna_analiza)" title="Pol (kompleksna analiza) – Slovenian" lang="sl" hreflang="sl" data-title="Pol (kompleksna analiza)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a 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class="sidebar-title-with-pretitle"><a href="/wiki/Complex_analysis" title="Complex analysis">Complex analysis</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Gamma_abs_3D.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Gamma_abs_3D.png/200px-Gamma_abs_3D.png" decoding="async" width="200" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Gamma_abs_3D.png/300px-Gamma_abs_3D.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Gamma_abs_3D.png/400px-Gamma_abs_3D.png 2x" data-file-width="1280" data-file-height="1000" /></a></span></td></tr><tr><th class="sidebar-heading"> <a href="/wiki/Complex_number" title="Complex number">Complex numbers</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Real_number" title="Real number">Real number</a></li> <li><a href="/wiki/Imaginary_number" title="Imaginary number">Imaginary number</a></li> <li><a href="/wiki/Complex_plane" title="Complex plane">Complex plane</a></li> <li><a href="/wiki/Complex_conjugate" title="Complex conjugate">Complex conjugate</a></li> <li><a href="/wiki/Unit_complex_number" class="mw-redirect" title="Unit complex number">Unit complex number</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Complex_analysis#Complex_functions" title="Complex analysis">Complex functions</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Complex_analysis#Complex_functions" title="Complex analysis">Complex-valued function</a></li> <li><a href="/wiki/Analytic_function" title="Analytic function">Analytic function</a></li> <li><a href="/wiki/Holomorphic_function" title="Holomorphic function">Holomorphic function</a></li> <li><a href="/wiki/Cauchy%E2%80%93Riemann_equations" title="Cauchy–Riemann equations">Cauchy–Riemann equations</a></li> <li><a href="/wiki/Formal_power_series" title="Formal power series">Formal power series</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Complex_analysis#Major_results" title="Complex analysis">Basic theory</a></th></tr><tr><td class="sidebar-content"> <ul><li><a class="mw-selflink selflink">Zeros and poles</a></li> <li><a href="/wiki/Cauchy%27s_integral_theorem" title="Cauchy's integral theorem">Cauchy's integral theorem</a></li> <li><a href="/wiki/Antiderivative_(complex_analysis)" title="Antiderivative (complex analysis)">Local primitive</a></li> <li><a href="/wiki/Cauchy%27s_integral_formula" title="Cauchy's integral formula">Cauchy's integral formula</a></li> <li><a href="/wiki/Winding_number" title="Winding number">Winding number</a></li> <li><a href="/wiki/Laurent_series" title="Laurent series">Laurent series</a></li> <li><a href="/wiki/Isolated_singularity" title="Isolated singularity">Isolated singularity</a></li> <li><a href="/wiki/Residue_theorem" title="Residue theorem">Residue theorem</a></li> <li><a href="/wiki/Argument_principle" title="Argument principle">Argument principle</a></li> <li><a href="/wiki/Conformal_map" title="Conformal map">Conformal map</a></li> <li><a href="/wiki/Schwarz_lemma" title="Schwarz lemma">Schwarz lemma</a></li> <li><a href="/wiki/Harmonic_function" title="Harmonic function">Harmonic function</a></li> <li><a href="/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace's equation</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Geometric_function_theory" title="Geometric function theory">Geometric function theory</a></th></tr><tr><th class="sidebar-heading"> <a href="/wiki/Complex_analysis#History" title="Complex analysis">People</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li> <li><a href="/wiki/Jacques_Hadamard" title="Jacques Hadamard">Jacques Hadamard</a></li> <li><a href="/wiki/Kiyoshi_Oka" title="Kiyoshi Oka">Kiyoshi Oka</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></li> <li><a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a></li></ul></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, 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a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Complex_analysis_sidebar" title="Template:Complex analysis sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Complex_analysis_sidebar" title="Template talk:Complex analysis sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Complex_analysis_sidebar" title="Special:EditPage/Template:Complex analysis sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a> (a branch of mathematics), a <b>pole</b> is a certain type of <a href="/wiki/Singularity_(mathematics)" title="Singularity (mathematics)">singularity</a> of a <a href="/wiki/Complex-valued_function" class="mw-redirect" title="Complex-valued function">complex-valued function</a> of a <a href="/wiki/Complex_number" title="Complex number">complex</a> variable. It is the simplest type of non-<a href="/wiki/Removable_singularity" title="Removable singularity">removable singularity</a> of such a function (see <a href="/wiki/Essential_singularity" title="Essential singularity">essential singularity</a>). Technically, a point <span class="texhtml"><i>z</i><sub>0</sub></span> is a pole of a function <span class="texhtml mvar" style="font-style:italic;">f</span> if it is a <a href="/wiki/Zero_of_a_function" title="Zero of a function">zero</a> of the function <span class="texhtml">1/<i>f</i></span> and <span class="texhtml">1/<i>f</i></span> is <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic</a> (i.e. <a href="/wiki/Complex_differentiable" class="mw-redirect" title="Complex differentiable">complex differentiable</a>) in some <a href="/wiki/Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighbourhood</a> of <span class="texhtml"><i>z</i><sub>0</sub></span>. </p><p>A function <span class="texhtml mvar" style="font-style:italic;">f</span> is <a href="/wiki/Meromorphic_function" title="Meromorphic function">meromorphic</a> in an <a href="/wiki/Open_set" title="Open set">open set</a> <span class="texhtml mvar" style="font-style:italic;">U</span> if for every point <span class="texhtml mvar" style="font-style:italic;">z</span> of <span class="texhtml mvar" style="font-style:italic;">U</span> there is a neighborhood of <span class="texhtml mvar" style="font-style:italic;">z</span> in which at least one of <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml">1/<i>f</i></span> is holomorphic. </p><p>If <span class="texhtml mvar" style="font-style:italic;">f</span> is meromorphic in <span class="texhtml mvar" style="font-style:italic;">U</span>, then a zero of <span class="texhtml mvar" style="font-style:italic;">f</span> is a pole of <span class="texhtml">1/<i>f</i></span>, and a pole of <span class="texhtml mvar" style="font-style:italic;">f</span> is a zero of <span class="texhtml">1/<i>f</i></span>. This induces a duality between <i>zeros</i> and <i>poles</i>, that is fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> plus the <a href="/wiki/Point_at_infinity" title="Point at infinity">point at infinity</a>, then the sum of the <a href="/wiki/Multiplicity_(mathematics)" title="Multiplicity (mathematics)">multiplicities</a> of its poles equals the sum of the multiplicities of its zeros. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeros_and_poles&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Function_of_a_complex_variable" class="mw-redirect" title="Function of a complex variable">function of a complex variable</a> <span class="texhtml mvar" style="font-style:italic;">z</span> is <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic</a> in an <a href="/wiki/Open_set" title="Open set">open domain</a> <span class="texhtml mvar" style="font-style:italic;">U</span> if it is <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable</a> with respect to <span class="texhtml mvar" style="font-style:italic;">z</span> at every point of <span class="texhtml mvar" style="font-style:italic;">U</span>. Equivalently, it is holomorphic if it is <a href="/wiki/Analytic_function" title="Analytic function">analytic</a>, that is, if its <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> exists at every point of <span class="texhtml mvar" style="font-style:italic;">U</span>, and converges to the function in some <a href="/wiki/Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighbourhood</a> of the point. A function is <a href="/wiki/Meromorphic_function" title="Meromorphic function">meromorphic</a> in <span class="texhtml mvar" style="font-style:italic;">U</span> if every point of <span class="texhtml mvar" style="font-style:italic;">U</span> has a neighbourhood such that at least one of <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml">1/<i>f</i></span> is holomorphic in it. </p><p>A <b><a href="/wiki/Zero_of_a_function" title="Zero of a function">zero</a></b> of a meromorphic function <span class="texhtml mvar" style="font-style:italic;">f</span> is a complex number <span class="texhtml mvar" style="font-style:italic;">z</span> such that <span class="texhtml"><i>f</i>(<i>z</i>) = 0</span>. A <b>pole</b> of <span class="texhtml mvar" style="font-style:italic;">f</span> is a zero of <span class="texhtml">1/<i>f</i></span>. </p><p>If <span class="texhtml mvar" style="font-style:italic;">f</span> is a function that is meromorphic in a neighbourhood of a point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e72d1d86e86355892b39b8eb32b964834e113bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}"></span> of the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>, then there exists an integer <span class="texhtml mvar" style="font-style:italic;">n</span> such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (z-z_{0})^{n}f(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (z-z_{0})^{n}f(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2a1dec9f088583e688684120ceddb33c7589601" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.268ex; height:2.843ex;" alt="{\displaystyle (z-z_{0})^{n}f(z)}"></span></dd></dl> <p>is holomorphic and nonzero in a neighbourhood of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e72d1d86e86355892b39b8eb32b964834e113bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}"></span> (this is a consequence of the analytic property). If <span class="texhtml"><i>n</i> > 0</span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e72d1d86e86355892b39b8eb32b964834e113bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}"></span> is a <i>pole</i> of <b>order</b> (or multiplicity) <span class="texhtml mvar" style="font-style:italic;">n</span> of <span class="texhtml mvar" style="font-style:italic;">f</span>. If <span class="texhtml"><i>n</i> < 0</span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e72d1d86e86355892b39b8eb32b964834e113bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}"></span> is a zero of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |n|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |n|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a62139afd28f74d3306e3bf603bebdecefe169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.688ex; height:2.843ex;" alt="{\displaystyle |n|}"></span> of <span class="texhtml mvar" style="font-style:italic;">f</span>. <i>Simple zero</i> and <i>simple pole</i> are terms used for zeroes and poles of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |n|=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |n|=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1756abfb9a55462af34cb24330e030e43d95861" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.596ex; height:2.843ex;" alt="{\displaystyle |n|=1.}"></span> <i>Degree</i> is sometimes used synonymously to order. </p><p>This characterization of zeros and poles implies that zeros and poles are <a href="/wiki/Isolated_point" title="Isolated point">isolated</a>, that is, every zero or pole has a neighbourhood that does not contain any other zero and pole. </p><p>Because of the <i>order</i> of zeros and poles being defined as a non-negative number <span class="texhtml mvar" style="font-style:italic;">n</span> and the symmetry between them, it is often useful to consider a pole of order <span class="texhtml mvar" style="font-style:italic;">n</span> as a zero of order <span class="texhtml">–<i>n</i></span> and a zero of order <span class="texhtml mvar" style="font-style:italic;">n</span> as a pole of order <span class="texhtml">–<i>n</i></span>. In this case a point that is neither a pole nor a zero is viewed as a pole (or zero) of order 0. </p><p>A meromorphic function may have infinitely many zeros and poles. This is the case for the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a> (see the image in the infobox), which is meromorphic in the whole complex plane, and has a simple pole at every non-positive integer. The <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> is also meromorphic in the whole complex plane, with a single pole of order 1 at <span class="texhtml"><i>z</i> = 1</span>. Its zeros in the left halfplane are all the negative even integers, and the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> is the conjecture that all other zeros are along <span class="texhtml">Re(<i>z</i>) = 1/2</span>. </p><p>In a neighbourhood of a point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b563889e721031251cd54f1a9fb6d04ed1a2e246" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.782ex; height:2.009ex;" alt="{\displaystyle z_{0},}"></span> a nonzero meromorphic function <span class="texhtml mvar" style="font-style:italic;">f</span> is the sum of a <a href="/wiki/Laurent_series" title="Laurent series">Laurent series</a> with at most finite <i>principal part</i> (the terms with negative index values): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=\sum _{k\geq -n}a_{k}(z-z_{0})^{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>≥</mo> <mo>−</mo> <mi>n</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)=\sum _{k\geq -n}a_{k}(z-z_{0})^{k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e498f49a9094bf68c951f80a7357280680498f5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.989ex; height:5.843ex;" alt="{\displaystyle f(z)=\sum _{k\geq -n}a_{k}(z-z_{0})^{k},}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">n</span> is an integer, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{-n}\neq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{-n}\neq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38cb5005d5b5a26504531f013527303bd6fb33fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.635ex; height:2.676ex;" alt="{\displaystyle a_{-n}\neq 0.}"></span> Again, if <span class="texhtml"><i>n</i> > 0</span> (the sum starts with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{-|n|}(z-z_{0})^{-|n|}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{-|n|}(z-z_{0})^{-|n|}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0b7b6f9cbc66f25eb46715999743805aec7de83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.927ex; height:3.676ex;" alt="{\displaystyle a_{-|n|}(z-z_{0})^{-|n|}}"></span>, the principal part has <span class="texhtml mvar" style="font-style:italic;">n</span> terms), one has a pole of order <span class="texhtml mvar" style="font-style:italic;">n</span>, and if <span class="texhtml"><i>n</i> ≤ 0</span> (the sum starts with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{|n|}(z-z_{0})^{|n|}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{|n|}(z-z_{0})^{|n|}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aaa23f85b2d48b641913752c82f57bd4ec3995e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.37ex; height:3.676ex;" alt="{\displaystyle a_{|n|}(z-z_{0})^{|n|}}"></span>, there is no principal part), one has a zero of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |n|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |n|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a62139afd28f74d3306e3bf603bebdecefe169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.688ex; height:2.843ex;" alt="{\displaystyle |n|}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="At_infinity">At infinity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeros_and_poles&action=edit&section=2" title="Edit section: At infinity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\mapsto f(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\mapsto f(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f73ee2be36803283b51b2916a7cb1f5082a23964" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.878ex; height:2.843ex;" alt="{\displaystyle z\mapsto f(z)}"></span> is <i>meromorphic at infinity</i> if it is meromorphic in some neighbourhood of infinity (that is outside some <a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">disk</a>), and there is an integer <span class="texhtml mvar" style="font-style:italic;">n</span> such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{z\to \infty }{\frac {f(z)}{z^{n}}}}"> <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>z</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{z\to \infty }{\frac {f(z)}{z^{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7af8e8bab36d45877fcbe573a04b831f7e3f649" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.455ex; height:5.676ex;" alt="{\displaystyle \lim _{z\to \infty }{\frac {f(z)}{z^{n}}}}"></span></dd></dl> <p>exists and is a nonzero complex number. </p><p>In this case, the <a href="/wiki/Point_at_infinity" title="Point at infinity">point at infinity</a> is a pole of order <span class="texhtml mvar" style="font-style:italic;">n</span> if <span class="texhtml"><i>n</i> > 0</span>, and a zero of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |n|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |n|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a62139afd28f74d3306e3bf603bebdecefe169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.688ex; height:2.843ex;" alt="{\displaystyle |n|}"></span> if <span class="texhtml"><i>n</i> < 0</span>. </p><p>For example, a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> of degree <span class="texhtml mvar" style="font-style:italic;">n</span> has a pole of degree <span class="texhtml mvar" style="font-style:italic;">n</span> at infinity. </p><p>The <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> extended by a point at infinity is called the <a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a>. </p><p>If <span class="texhtml mvar" style="font-style:italic;">f</span> is a function that is meromorphic on the whole Riemann sphere, then it has a finite number of zeros and poles, and the sum of the orders of its poles equals the sum of the orders of its zeros. </p><p>Every <a href="/wiki/Rational_function" title="Rational function">rational function</a> is meromorphic on the whole Riemann sphere, and, in this case, the sum of orders of the zeros or of the poles is the maximum of the degrees of the numerator and the denominator. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeros_and_poles&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Pole-order9-infin.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Pole-order9-infin.png/300px-Pole-order9-infin.png" decoding="async" width="300" height="273" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Pole-order9-infin.png/450px-Pole-order9-infin.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/75/Pole-order9-infin.png 2x" data-file-width="558" data-file-height="508" /></a><figcaption>A polynomial of degree 9 has a pole of order 9 at ∞, here plotted by <a href="/wiki/Domain_coloring" title="Domain coloring">domain coloring</a> of the Riemann sphere.</figcaption></figure> <ul><li>The function</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)={\frac {3}{z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mi>z</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)={\frac {3}{z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0f99277c10ca8824079712160c269643e79231a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.273ex; height:5.176ex;" alt="{\displaystyle f(z)={\frac {3}{z}}}"></span></dd></dl></dd> <dd>is meromorphic on the whole Riemann sphere. It has a pole of order 1 or simple pole at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/463359fa7c7563dc29f2079e63195b0035f1ab5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.996ex; height:2.509ex;" alt="{\displaystyle z=0,}"></span> and a simple zero at infinity.</dd></dl> <ul><li>The function</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)={\frac {z+2}{(z-5)^{2}(z+7)^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>+</mo> <mn>2</mn> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>5</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mn>7</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)={\frac {z+2}{(z-5)^{2}(z+7)^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a731733dea44dd378e509c51bbf187b51e5b68d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.02ex; height:6.009ex;" alt="{\displaystyle f(z)={\frac {z+2}{(z-5)^{2}(z+7)^{3}}}}"></span></dd></dl></dd> <dd>is meromorphic on the whole Riemann sphere. It has a pole of order 2 at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=5,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=5,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed24ac1b45bfec32523ce4128a1a190cea93eb8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.996ex; height:2.509ex;" alt="{\displaystyle z=5,}"></span> and a pole of order 3 at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=-7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=-7}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dbc650f6bba21c7a94cb8222b6f457593f8e4fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.157ex; height:2.343ex;" alt="{\displaystyle z=-7}"></span>. It has a simple zero at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=-2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=-2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f919665c4e6b29bd170e3edb78feb47ea861f06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.804ex; height:2.509ex;" alt="{\displaystyle z=-2,}"></span> and a quadruple zero at infinity.</dd></dl> <ul><li>The function</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)={\frac {z-4}{e^{z}-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>−</mo> <mn>4</mn> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)={\frac {z-4}{e^{z}-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc82d869377e3ac1616edd7c68de6b567b24576" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.199ex; height:5.343ex;" alt="{\displaystyle f(z)={\frac {z-4}{e^{z}-1}}}"></span></dd></dl></dd> <dd>is meromorphic in the whole complex plane, but not at infinity. It has poles of order 1 at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=2\pi ni{\text{ for }}n\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=2\pi ni{\text{ for }}n\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b308a37bc37a2a43d88b8e0df55f801d57f97be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.611ex; height:2.176ex;" alt="{\displaystyle z=2\pi ni{\text{ for }}n\in \mathbb {Z} }"></span>. This can be seen by writing the <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4772def31b56e642df3e4d1160cadff3d80ba45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.085ex; height:2.343ex;" alt="{\displaystyle e^{z}}"></span> around the origin.</dd></dl> <ul><li>The function</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)=z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a47d2cd0ea040d1b1af7532f5321ed581026e61f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.363ex; height:2.843ex;" alt="{\displaystyle f(z)=z}"></span></dd></dl></dd> <dd>has a single pole at infinity of order 1, and a single zero at the origin.</dd></dl> <p>All above examples except for the third are <a href="/wiki/Rational_functions" class="mw-redirect" title="Rational functions">rational functions</a>. For a general discussion of zeros and poles of such functions, see <a href="/wiki/Pole%E2%80%93zero_plot#Continuous-time_systems" title="Pole–zero plot">Pole–zero plot § Continuous-time systems</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Function_on_a_curve">Function on a curve</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeros_and_poles&action=edit&section=4" title="Edit section: Function on a curve"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The concept of zeros and poles extends naturally to functions on a <i>complex curve</i>, that is <a href="/wiki/Complex_analytic_manifold" class="mw-redirect" title="Complex analytic manifold">complex analytic manifold</a> of dimension one (over the complex numbers). The simplest examples of such curves are the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> and the <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surface</a>. This extension is done by transferring structures and properties through <a href="/wiki/Atlas_(topology)" title="Atlas (topology)">charts</a>, which are analytic <a href="/wiki/Isomorphism" title="Isomorphism">isomorphisms</a>. </p><p>More precisely, let <span class="texhtml mvar" style="font-style:italic;">f</span> be a function from a complex curve <span class="texhtml mvar" style="font-style:italic;">M</span> to the complex numbers. This function is holomorphic (resp. meromorphic) in a neighbourhood of a point <span class="texhtml mvar" style="font-style:italic;">z</span> of <span class="texhtml mvar" style="font-style:italic;">M</span> if there is a chart <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ \phi ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ \phi ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa227c8abb67f2401820a4c4f4931173448dee5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.192ex; height:3.009ex;" alt="{\displaystyle f\circ \phi ^{-1}}"></span> is holomorphic (resp. meromorphic) in a neighbourhood of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (z).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (z).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eac2d526fa25956b78fe92988531c68e3a2ffe7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.93ex; height:2.843ex;" alt="{\displaystyle \phi (z).}"></span> Then, <span class="texhtml mvar" style="font-style:italic;">z</span> is a pole or a zero of order <span class="texhtml mvar" style="font-style:italic;">n</span> if the same is true for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (z).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (z).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eac2d526fa25956b78fe92988531c68e3a2ffe7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.93ex; height:2.843ex;" alt="{\displaystyle \phi (z).}"></span> </p><p>If the curve is <a href="/wiki/Compact_space" title="Compact space">compact</a>, and the function <span class="texhtml mvar" style="font-style:italic;">f</span> is meromorphic on the whole curve, then the number of zeros and poles is finite, and the sum of the orders of the poles equals the sum of the orders of the zeros. This is one of the basic facts that are involved in <a href="/wiki/Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeros_and_poles&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Argument_principle" title="Argument principle">Argument principle</a></li> <li><a href="/wiki/Control_theory#Stability" title="Control theory">Control theory § Stability</a></li> <li><a href="/wiki/Filter_design" title="Filter design">Filter design</a></li> <li><a href="/wiki/Filter_(signal_processing)" title="Filter (signal processing)">Filter (signal processing)</a></li> <li><a href="/wiki/Gauss%E2%80%93Lucas_theorem" title="Gauss–Lucas theorem">Gauss–Lucas theorem</a></li> <li><a href="/wiki/Hurwitz%27s_theorem_(complex_analysis)" title="Hurwitz's theorem (complex analysis)">Hurwitz's theorem (complex analysis)</a></li> <li><a href="/wiki/Marden%27s_theorem" title="Marden's theorem">Marden's theorem</a></li> <li><a href="/wiki/Nyquist_stability_criterion" title="Nyquist stability criterion">Nyquist stability criterion</a></li> <li><a href="/wiki/Pole%E2%80%93zero_plot" title="Pole–zero plot">Pole–zero plot</a></li> <li><a href="/wiki/Residue_(complex_analysis)" title="Residue (complex analysis)">Residue (complex analysis)</a></li> <li><a href="/wiki/Rouch%C3%A9%27s_theorem" title="Rouché's theorem">Rouché's theorem</a></li> <li><a href="/wiki/Sendov%27s_conjecture" title="Sendov's conjecture">Sendov's conjecture</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeros_and_poles&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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="CITEREFConway1986" class="citation book cs1"><a href="/wiki/John_B._Conway" title="John B. Conway">Conway, John B.</a> (1986). <i>Functions of One Complex Variable I</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90328-3" title="Special:BookSources/0-387-90328-3"><bdi>0-387-90328-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functions+of+One+Complex+Variable+I&rft.pub=Springer&rft.date=1986&rft.isbn=0-387-90328-3&rft.aulast=Conway&rft.aufirst=John+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeros+and+poles" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConway1995" class="citation book cs1">Conway, John B. (1995). <i>Functions of One Complex Variable II</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94460-5" title="Special:BookSources/0-387-94460-5"><bdi>0-387-94460-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functions+of+One+Complex+Variable+II&rft.pub=Springer&rft.date=1995&rft.isbn=0-387-94460-5&rft.aulast=Conway&rft.aufirst=John+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeros+and+poles" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHenrici1974" class="citation book cs1"><a href="/wiki/Peter_Henrici_(mathematician)" title="Peter Henrici (mathematician)">Henrici, Peter</a> (1974). <i>Applied and Computational Complex Analysis 1</i>. <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+and+Computational+Complex+Analysis+1&rft.pub=John+Wiley+%26+Sons&rft.date=1974&rft.aulast=Henrici&rft.aufirst=Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeros+and+poles" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeros_and_poles&action=edit&section=7" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Pole"><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/Pole.html">"Pole"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</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=Pole&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FPole.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeros+and+poles" class="Z3988"></span></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Zeros_and_poles&oldid=1229392443">https://en.wikipedia.org/w/index.php?title=Zeros_and_poles&oldid=1229392443</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Complex_analysis" title="Category:Complex analysis">Complex analysis</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 16 June 2024, at 15:16<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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