Gudermannian function

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Available in 19 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-19" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">19 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%A7%D9%84%D8%A9_%D8%BA%D9%88%D8%AF%D8%B1%D9%85%D8%A7%D9%86%D9%8A%D8%A9" title="دالة غودرمانية – Arabic" lang="ar" hreflang="ar" data-title="دالة غودرمانية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Gudermannova_funkcija" title="Gudermannova funkcija – Bosnian" lang="bs" hreflang="bs" data-title="Gudermannova funkcija" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3_gudermanniana" title="Funció gudermanniana – Catalan" lang="ca" hreflang="ca" data-title="Funció gudermanniana" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gudermannfunktion" title="Gudermannfunktion – German" lang="de" hreflang="de" data-title="Gudermannfunktion" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_de_Gudermann" title="Función de Gudermann – Spanish" lang="es" hreflang="es" data-title="Función de Gudermann" 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/Funkcio_de_Gudermannian" title="Funkcio de Gudermannian – Esperanto" lang="eo" hreflang="eo" data-title="Funkcio de Gudermannian" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Fonction_de_Gudermann" title="Fonction de Gudermann – French" lang="fr" hreflang="fr" data-title="Fonction de Gudermann" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_gudermanniana" title="Funzione gudermanniana – Italian" lang="it" hreflang="it" data-title="Funzione gudermanniana" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Gudermann-f%C3%BCggv%C3%A9ny" title="Gudermann-függvény – Hungarian" lang="hu" hreflang="hu" data-title="Gudermann-függvény" 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/Gudermannfunctie" title="Gudermannfunctie – Dutch" lang="nl" hreflang="nl" data-title="Gudermannfunctie" 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/%E3%82%B0%E3%83%BC%E3%83%87%E3%83%AB%E3%83%9E%E3%83%B3%E9%96%A2%E6%95%B0" title="グーデルマン関数 – Japanese" lang="ja" hreflang="ja" data-title="グーデルマン関数" 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-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%A2%E1%9E%93%E1%9E%BB%E1%9E%82%E1%9E%98%E1%9E%93%E1%9F%8D%E1%9E%A0%E1%9F%92%E1%9E%82%E1%9E%BB%E1%9E%8C%E1%9F%82%E1%9E%9A%E1%9E%98%E1%9F%89%E1%9E%B6%E1%9E%93%E1%9F%8B" title="អនុគមន៍ហ្គុឌែរម៉ាន់ – Khmer" lang="km" hreflang="km" data-title="អនុគមន៍ហ្គុឌែរម៉ាន់" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcja_Gudermanna" title="Funkcja Gudermanna – Polish" lang="pl" hreflang="pl" data-title="Funkcja Gudermanna" 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/Fun%C3%A7%C3%A3o_gudermanniana" title="Função gudermanniana – Portuguese" lang="pt" hreflang="pt" data-title="Função gudermanniana" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D0%93%D1%83%D0%B4%D0%B5%D1%80%D0%BC%D0%B0%D0%BD%D0%B0" title="Функция Гудермана – Russian" lang="ru" hreflang="ru" data-title="Функция Гудермана" 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/Gudermannova_funkcija" title="Gudermannova funkcija – Slovenian" lang="sl" hreflang="sl" data-title="Gudermannova funkcija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Gudermannin_funktio" title="Gudermannin funktio – Finnish" lang="fi" hreflang="fi" data-title="Gudermannin funktio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%93%D1%83%D0%B4%D0%B5%D1%80%D0%BC%D0%B0%D0%BD%D0%B0" title="Функція Гудермана – Ukrainian" lang="uk" hreflang="uk" data-title="Функція Гудермана" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8F%A4%E5%BE%B7%E6%9B%BC%E5%87%BD%E6%95%B8" title="古德曼函數 – Chinese" lang="zh" hreflang="zh" data-title="古德曼函數" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1328149#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> 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hyperbolic functions</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gudermannian_function.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Gudermannian_function.png/330px-Gudermannian_function.png" decoding="async" width="330" height="254" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Gudermannian_function.png/495px-Gudermannian_function.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Gudermannian_function.png/660px-Gudermannian_function.png 2x" data-file-width="1652" data-file-height="1274" /></a><figcaption>The Gudermannian function relates the area of a <a href="/wiki/Circular_sector" title="Circular sector">circular sector</a> to the area of a <a href="/wiki/Hyperbolic_sector" title="Hyperbolic sector">hyperbolic sector</a>, via a common <a href="/wiki/Stereographic_projection" title="Stereographic projection">stereographic projection</a>. If twice the area of the blue hyperbolic sector is <span class="texhtml"><i>ψ</i></span>, then twice the area of the red circular sector is <span class="texhtml"><i>ϕ</i> = gd <i>ψ</i></span>. Twice the area of the purple triangle is the stereographic projection <span class="texhtml"><i>s</i> = tan <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><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>ϕ</i> = tanh <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>ψ</i>.</span> The blue point has coordinates <span class="texhtml">(cosh <i>ψ</i>, sinh <i>ψ</i>)</span>. The red point has coordinates <span class="texhtml">(cos <i>ϕ</i>, sin <i>ϕ</i>).</span> The purple point has coordinates <span class="texhtml">(0, <i>s</i>).</span> </figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Gudermannian_graph.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Gudermannian_graph.png/310px-Gudermannian_graph.png" decoding="async" width="310" height="189" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Gudermannian_graph.png/465px-Gudermannian_graph.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/Gudermannian_graph.png/620px-Gudermannian_graph.png 2x" data-file-width="1846" data-file-height="1128" /></a><figcaption><a href="/wiki/Graph_of_a_function" title="Graph of a function">Graph</a> of the Gudermannian function.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Inverse_Gudermannian_graph.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Inverse_Gudermannian_graph.png/260px-Inverse_Gudermannian_graph.png" decoding="async" width="260" height="231" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Inverse_Gudermannian_graph.png/390px-Inverse_Gudermannian_graph.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Inverse_Gudermannian_graph.png/520px-Inverse_Gudermannian_graph.png 2x" data-file-width="1590" data-file-height="1412" /></a><figcaption>Graph of the inverse Gudermannian function.</figcaption></figure> <p>In mathematics, the <b>Gudermannian function</b> relates a <a href="/wiki/Hyperbolic_angle" title="Hyperbolic angle">hyperbolic angle</a> measure <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="{\textstyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d738e3571903ec4e786923ddbd817cd147cb5a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\textstyle \psi }"></span> to a <a href="/wiki/Angle" title="Angle">circular angle</a> measure <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="{\textstyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b36dc4887c86d6f01183292a5d2f984e23318c10" 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="{\textstyle \phi }"></span> called the <i>gudermannian</i> 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="{\textstyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d738e3571903ec4e786923ddbd817cd147cb5a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\textstyle \psi }"></span> and denoted <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="{\textstyle \operatorname {gd} \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>gd</mi> <mo>⁡</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {gd} \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7833c345c8798d7f6a5b4401d71a3f1a6f8d7b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.355ex; height:2.509ex;" alt="{\textstyle \operatorname {gd} \psi }"></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The Gudermannian function reveals a close relationship between the <a href="/wiki/Circular_function" class="mw-redirect" title="Circular function">circular functions</a> and <a href="/wiki/Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">hyperbolic functions</a>. It was introduced in the 1760s by <a href="/wiki/Johann_Heinrich_Lambert" title="Johann Heinrich Lambert">Johann Heinrich Lambert</a>, and later named for <a href="/wiki/Christoph_Gudermann" title="Christoph Gudermann">Christoph Gudermann</a> who also described the relationship between circular and hyperbolic functions in 1830.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The gudermannian is sometimes called the <b>hyperbolic amplitude</b> as a <a href="/wiki/Limiting_case_(mathematics)" title="Limiting case (mathematics)">limiting case</a> of the <a href="/wiki/Jacobi_elliptic_functions#am" title="Jacobi elliptic functions">Jacobi elliptic amplitude</a> <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="{\textstyle \operatorname {am} (\psi ,m)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>am</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {am} (\psi ,m)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb75dec9c8aa0e577f2f558db0f83dae9f2ef77e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.495ex; height:2.843ex;" alt="{\textstyle \operatorname {am} (\psi ,m)}"></span> when parameter <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="{\textstyle m=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle m=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25d81740fd298793563fa19a8e0ed08fdd89738e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.948ex; height:2.176ex;" alt="{\textstyle m=1.}"></span> </p><p>The <a href="/wiki/Real_number" title="Real number">real</a> Gudermannian function is typically defined 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="{\textstyle -\infty <\psi <\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo><</mo> <mi>ψ</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -\infty <\psi <\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e75be30118b3e1539d58c04f169ac31b90ed3a29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.166ex; height:2.509ex;" alt="{\textstyle -\infty <\psi <\infty }"></span> to be the integral of the hyperbolic secant<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </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 \phi =\operatorname {gd} \psi \equiv \int _{0}^{\psi }\operatorname {sech} t\,\mathrm {d} t=\operatorname {arctan} (\sinh \psi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mi>gd</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>≡</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msubsup> <mi>sech</mi> <mo>⁡</mo> <mi>t</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sinh</mi> <mo>⁡</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =\operatorname {gd} \psi \equiv \int _{0}^{\psi }\operatorname {sech} t\,\mathrm {d} t=\operatorname {arctan} (\sinh \psi ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bda4b2ebdd63e0a668c77c86cf2075b2effa2832" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.18ex; height:6.343ex;" alt="{\displaystyle \phi =\operatorname {gd} \psi \equiv \int _{0}^{\psi }\operatorname {sech} t\,\mathrm {d} t=\operatorname {arctan} (\sinh \psi ).}"></span></dd></dl> <p>The real inverse Gudermannian function can be defined 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="{\textstyle -{\tfrac {1}{2}}\pi <\phi <{\tfrac {1}{2}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo><</mo> <mi>ϕ</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -{\tfrac {1}{2}}\pi <\phi <{\tfrac {1}{2}}\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3ae0f4df0531eaba3c0a793974483285508471" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.371ex; height:3.509ex;" alt="{\textstyle -{\tfrac {1}{2}}\pi <\phi <{\tfrac {1}{2}}\pi }"></span> as the <a href="/wiki/Integral_of_the_secant_function" title="Integral of the secant function">integral of the (circular) secant</a> </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 \psi =\operatorname {gd} ^{-1}\phi =\int _{0}^{\phi }\operatorname {sec} t\,\mathrm {d} t=\operatorname {arsinh} (\tan \phi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msubsup> <mi>sec</mi> <mo>⁡</mo> <mi>t</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mi>arsinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>tan</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =\operatorname {gd} ^{-1}\phi =\int _{0}^{\phi }\operatorname {sec} t\,\mathrm {d} t=\operatorname {arsinh} (\tan \phi ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76c0302e0820820cf3804ab0b839f12d330716d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.969ex; height:6.343ex;" alt="{\displaystyle \psi =\operatorname {gd} ^{-1}\phi =\int _{0}^{\phi }\operatorname {sec} t\,\mathrm {d} t=\operatorname {arsinh} (\tan \phi ).}"></span></dd></dl> <p>The hyperbolic angle measure <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 \psi =\operatorname {gd} ^{-1}\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =\operatorname {gd} ^{-1}\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82961cdf95e4e7a4f3b2077f565baf68b60742dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.172ex; height:3.009ex;" alt="{\displaystyle \psi =\operatorname {gd} ^{-1}\phi }"></span> is called the <i>anti-gudermannian</i> 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 }"> <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> or sometimes the <b>lambertian</b> 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 }"> <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>, denoted <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 \psi =\operatorname {lam} \phi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mi>lam</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =\operatorname {lam} \phi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/430b2e5e131383a2f2c52c06f67f82588bb4a61d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.776ex; height:2.509ex;" alt="{\displaystyle \psi =\operatorname {lam} \phi .}"></span><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In the context of <a href="/wiki/Geodesy" title="Geodesy">geodesy</a> and <a href="/wiki/Navigation" title="Navigation">navigation</a> for latitude <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="{\textstyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b36dc4887c86d6f01183292a5d2f984e23318c10" 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="{\textstyle \phi }"></span>, <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 k\operatorname {gd} ^{-1}\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\operatorname {gd} ^{-1}\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9a654a3f871e362330c6c00df4dcf088bdb4143" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.159ex; height:3.009ex;" alt="{\displaystyle k\operatorname {gd} ^{-1}\phi }"></span> (scaled by arbitrary constant <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="{\textstyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5595fc0c47452f8fc2aa6e29c3611f047714b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\textstyle k}"></span>) was historically called the <b>meridional part</b> 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 }"> <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> (<a href="/wiki/French_(language)" class="mw-redirect" title="French (language)">French</a>: <i>latitude croissante</i>). It is the vertical coordinate of the <a href="/wiki/Mercator_projection" title="Mercator projection">Mercator projection</a>. </p><p>The two angle measures <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="{\textstyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b36dc4887c86d6f01183292a5d2f984e23318c10" 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="{\textstyle \phi }"></span> 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="{\textstyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d738e3571903ec4e786923ddbd817cd147cb5a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\textstyle \psi }"></span> are related by a common <a href="/wiki/Stereographic_projection" title="Stereographic projection">stereographic projection</a> </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 s=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>ϕ</mi> <mo>=</mo> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>ψ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dea5fd46ee49cac81ff9bd5960d1e46ecddaad1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.935ex; height:3.509ex;" alt="{\displaystyle s=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi ,}"></span></dd></dl> <p>and this identity can serve as an alternative definition 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="{\textstyle \operatorname {gd} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>gd</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {gd} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e8496c6ca1328821c3f42dd1173a63d6dacbd05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\textstyle \operatorname {gd} }"></span> 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="{\textstyle \operatorname {gd} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {gd} ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c67c70418c886a7bc07a9078fecbef157b393a99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.788ex; height:3.009ex;" alt="{\textstyle \operatorname {gd} ^{-1}}"></span> valid throughout the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>: </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 {\begin{aligned}\operatorname {gd} \psi &={2\arctan }{\bigl (}\tanh {\tfrac {1}{2}}\psi \,{\bigr )},\\[5mu]\operatorname {gd} ^{-1}\phi &={2\operatorname {artanh} }{\bigl (}\tan {\tfrac {1}{2}}\phi \,{\bigr )}.\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="0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>gd</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>arctan</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>ψ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>artanh</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>ϕ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {gd} \psi &={2\arctan }{\bigl (}\tanh {\tfrac {1}{2}}\psi \,{\bigr )},\\[5mu]\operatorname {gd} ^{-1}\phi &={2\operatorname {artanh} }{\bigl (}\tan {\tfrac {1}{2}}\phi \,{\bigr )}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dd27bd5d8e880b8a5a6bdeff8d4a27df40264c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:30.187ex; height:8.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {gd} \psi &={2\arctan }{\bigl (}\tanh {\tfrac {1}{2}}\psi \,{\bigr )},\\[5mu]\operatorname {gd} ^{-1}\phi &={2\operatorname {artanh} }{\bigl (}\tan {\tfrac {1}{2}}\phi \,{\bigr )}.\end{aligned}}}"></span></dd></dl> <meta property="mw:PageProp/toc" /> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Circular–hyperbolic_identities"><span id="Circular.E2.80.93hyperbolic_identities"></span>Circular–hyperbolic identities</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gudermannian_function&action=edit&section=1" title="Edit section: Circular–hyperbolic identities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We can evaluate the integral of the hyperbolic secant using the stereographic projection (<a href="/wiki/Tangent_half-angle_substitution#Hyperbolic_functions" title="Tangent half-angle substitution">hyperbolic half-tangent</a>) as a <a href="/wiki/Integration_by_substitution" title="Integration by substitution">change of variables</a>:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </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 {\begin{aligned}\operatorname {gd} \psi &\equiv \int _{0}^{\psi }{\frac {1}{\operatorname {cosh} t}}\mathrm {d} t=\int _{0}^{\tanh {\frac {1}{2}}\psi }{\frac {1-u^{2}}{1+u^{2}}}{\frac {2\,\mathrm {d} u}{1-u^{2}}}\qquad {\bigl (}u=\tanh {\tfrac {1}{2}}t{\bigr )}\\[8mu]&=2\int _{0}^{\tanh {\frac {1}{2}}\psi }{\frac {1}{1+u^{2}}}\mathrm {d} u={2\arctan }{\bigl (}\tanh {\tfrac {1}{2}}\psi \,{\bigr )},\\[5mu]\tan {\tfrac {1}{2}}{\operatorname {gd} \psi }&=\tanh {\tfrac {1}{2}}\psi .\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="0.744em 0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>gd</mi> <mo>⁡</mo> <mi>ψ</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>ψ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cosh</mi> <mo>⁡</mo> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ψ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>u</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>u</mi> <mo>=</mo> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ψ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>arctan</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>ψ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>gd</mi> <mo>⁡</mo> <mi>ψ</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>ψ</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {gd} \psi &\equiv \int _{0}^{\psi }{\frac {1}{\operatorname {cosh} t}}\mathrm {d} t=\int _{0}^{\tanh {\frac {1}{2}}\psi }{\frac {1-u^{2}}{1+u^{2}}}{\frac {2\,\mathrm {d} u}{1-u^{2}}}\qquad {\bigl (}u=\tanh {\tfrac {1}{2}}t{\bigr )}\\[8mu]&=2\int _{0}^{\tanh {\frac {1}{2}}\psi }{\frac {1}{1+u^{2}}}\mathrm {d} u={2\arctan }{\bigl (}\tanh {\tfrac {1}{2}}\psi \,{\bigr )},\\[5mu]\tan {\tfrac {1}{2}}{\operatorname {gd} \psi }&=\tanh {\tfrac {1}{2}}\psi .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5945db7bc55d0b033a77986e9cc3ea1c2c529eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.91ex; margin-bottom: -0.261ex; width:71.98ex; height:19.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {gd} \psi &\equiv \int _{0}^{\psi }{\frac {1}{\operatorname {cosh} t}}\mathrm {d} t=\int _{0}^{\tanh {\frac {1}{2}}\psi }{\frac {1-u^{2}}{1+u^{2}}}{\frac {2\,\mathrm {d} u}{1-u^{2}}}\qquad {\bigl (}u=\tanh {\tfrac {1}{2}}t{\bigr )}\\[8mu]&=2\int _{0}^{\tanh {\frac {1}{2}}\psi }{\frac {1}{1+u^{2}}}\mathrm {d} u={2\arctan }{\bigl (}\tanh {\tfrac {1}{2}}\psi \,{\bigr )},\\[5mu]\tan {\tfrac {1}{2}}{\operatorname {gd} \psi }&=\tanh {\tfrac {1}{2}}\psi .\end{aligned}}}"></span></dd></dl> <p>Letting <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="{\textstyle \phi =\operatorname {gd} \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mi>gd</mi> <mo>⁡</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi =\operatorname {gd} \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a90392c8dc8bdc35d51f9b99dd6d134d4300a4b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.839ex; height:2.509ex;" alt="{\textstyle \phi =\operatorname {gd} \psi }"></span> 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="{\textstyle s=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>ϕ</mi> <mo>=</mo> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle s=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8df0ddd190fa55c408b835fbccf2e6c73a0e3afc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.288ex; height:3.509ex;" alt="{\textstyle s=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi }"></span> we can derive a number of identities between hyperbolic functions 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="{\textstyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d738e3571903ec4e786923ddbd817cd147cb5a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\textstyle \psi }"></span> and circular functions 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="{\textstyle \phi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ϕ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0171a72f233f82af593e5c440d7b5cd3ae388af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.032ex; height:2.509ex;" alt="{\textstyle \phi .}"></span><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Frameless"><a href="/wiki/File:Gudermannian_identities.png" class="mw-file-description" title="Identities related to the Gudermannian function represented graphically."><img alt="Identities related to the Gudermannian function represented graphically." src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Gudermannian_identities.png/440px-Gudermannian_identities.png" decoding="async" width="440" height="357" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Gudermannian_identities.png/660px-Gudermannian_identities.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Gudermannian_identities.png/880px-Gudermannian_identities.png 2x" data-file-width="1510" data-file-height="1226" /></a><figcaption>Identities related to the Gudermannian function represented graphically.</figcaption></figure> <div style="clear:left;" class=""></div> <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 {\begin{aligned}s&=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi ,\\[6mu]{\frac {2s}{1+s^{2}}}&=\sin \phi =\tanh \psi ,\quad &{\frac {1+s^{2}}{2s}}&=\csc \phi =\coth \psi ,\\[10mu]{\frac {1-s^{2}}{1+s^{2}}}&=\cos \phi =\operatorname {sech} \psi ,\quad &{\frac {1+s^{2}}{1-s^{2}}}&=\sec \phi =\cosh \psi ,\\[10mu]{\frac {2s}{1-s^{2}}}&=\tan \phi =\sinh \psi ,\quad &{\frac {1-s^{2}}{2s}}&=\cot \phi =\operatorname {csch} \psi .\\[8mu]\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="0.633em 0.856em 0.856em 0.744em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>s</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>ϕ</mi> <mo>=</mo> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>ψ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>s</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>=</mo> <mi>tanh</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>,</mo> <mspace width="1em" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>s</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>csc</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>=</mo> <mi>coth</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>=</mo> <mi>sech</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>,</mo> <mspace width="1em" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sec</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>=</mo> <mi>cosh</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>s</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>=</mo> <mi>sinh</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>,</mo> <mspace width="1em" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>s</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cot</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>=</mo> <mi>csch</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s&=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi ,\\[6mu]{\frac {2s}{1+s^{2}}}&=\sin \phi =\tanh \psi ,\quad &{\frac {1+s^{2}}{2s}}&=\csc \phi =\coth \psi ,\\[10mu]{\frac {1-s^{2}}{1+s^{2}}}&=\cos \phi =\operatorname {sech} \psi ,\quad &{\frac {1+s^{2}}{1-s^{2}}}&=\sec \phi =\cosh \psi ,\\[10mu]{\frac {2s}{1-s^{2}}}&=\tan \phi =\sinh \psi ,\quad &{\frac {1-s^{2}}{2s}}&=\cot \phi =\operatorname {csch} \psi .\\[8mu]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b899ada2ea0fb5c5f68d29fefb9498e1d57d2ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:59.11ex; height:25.509ex;" alt="{\displaystyle {\begin{aligned}s&=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi ,\\[6mu]{\frac {2s}{1+s^{2}}}&=\sin \phi =\tanh \psi ,\quad &{\frac {1+s^{2}}{2s}}&=\csc \phi =\coth \psi ,\\[10mu]{\frac {1-s^{2}}{1+s^{2}}}&=\cos \phi =\operatorname {sech} \psi ,\quad &{\frac {1+s^{2}}{1-s^{2}}}&=\sec \phi =\cosh \psi ,\\[10mu]{\frac {2s}{1-s^{2}}}&=\tan \phi =\sinh \psi ,\quad &{\frac {1-s^{2}}{2s}}&=\cot \phi =\operatorname {csch} \psi .\\[8mu]\end{aligned}}}"></span></dd></dl> <p>These are commonly used as expressions 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 \operatorname {gd} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>gd</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gd} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dad73bd4edce09e79f274f0bfcd574c0c32a46a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \operatorname {gd} }"></span> 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 \operatorname {gd} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gd} ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a79e1a6ee5398ea160393c746ff96902f6857fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.788ex; height:3.009ex;" alt="{\displaystyle \operatorname {gd} ^{-1}}"></span> for real values 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 \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> 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 \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> 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 |\phi |<{\tfrac {1}{2}}\pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi |<{\tfrac {1}{2}}\pi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad50d61647f0c09b281efef44887ad39c5cdab78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.415ex; height:3.509ex;" alt="{\displaystyle |\phi |<{\tfrac {1}{2}}\pi .}"></span> For example, the numerically well-behaved formulas </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 {\begin{aligned}\operatorname {gd} \psi &=\operatorname {arctan} (\sinh \psi ),\\[6mu]\operatorname {gd} ^{-1}\phi &=\operatorname {arsinh} (\tan \phi ).\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="0.633em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>gd</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sinh</mi> <mo>⁡</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arsinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>tan</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {gd} \psi &=\operatorname {arctan} (\sinh \psi ),\\[6mu]\operatorname {gd} ^{-1}\phi &=\operatorname {arsinh} (\tan \phi ).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cbb02476a2f58f8303e30dde2cd910b3357ea87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.381ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {gd} \psi &=\operatorname {arctan} (\sinh \psi ),\\[6mu]\operatorname {gd} ^{-1}\phi &=\operatorname {arsinh} (\tan \phi ).\end{aligned}}}"></span></dd></dl> <p>(Note, 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 |>{\tfrac {1}{2}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi |>{\tfrac {1}{2}}\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ebf58c954a4cb4f5f6588cc1ae44649ff7c455b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.768ex; height:3.509ex;" alt="{\displaystyle |\phi |>{\tfrac {1}{2}}\pi }"></span> and for complex arguments, care must be taken choosing <a href="/wiki/Branch_point" title="Branch point">branches</a> of the inverse functions.)<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>We can also express <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="{\textstyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d738e3571903ec4e786923ddbd817cd147cb5a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\textstyle \psi }"></span> 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="{\textstyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b36dc4887c86d6f01183292a5d2f984e23318c10" 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="{\textstyle \phi }"></span> in terms 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="{\textstyle s\colon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>s</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle s\colon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f10a92e0bdd50eb65472974064e685ad708acd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.737ex; height:1.676ex;" alt="{\textstyle s\colon }"></span> </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 {\begin{aligned}2\arctan s&=\phi =\operatorname {gd} \psi ,\\[6mu]2\operatorname {artanh} s&=\operatorname {gd} ^{-1}\phi =\psi .\\[6mu]\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="0.633em 0.633em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mi>s</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ϕ</mi> <mo>=</mo> <mi>gd</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>artanh</mi> <mo>⁡</mo> <mi>s</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> <mo>=</mo> <mi>ψ</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}2\arctan s&=\phi =\operatorname {gd} \psi ,\\[6mu]2\operatorname {artanh} s&=\operatorname {gd} ^{-1}\phi =\psi .\\[6mu]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/430ca5043a59099968092d81866f11c67f7ee130" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.422ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}2\arctan s&=\phi =\operatorname {gd} \psi ,\\[6mu]2\operatorname {artanh} s&=\operatorname {gd} ^{-1}\phi =\psi .\\[6mu]\end{aligned}}}"></span></dd></dl> <p>If we expand <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="{\textstyle \tan {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \tan {\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14c70fad71d4650185650afdb63830356769a30f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.405ex; height:3.509ex;" alt="{\textstyle \tan {\tfrac {1}{2}}}"></span> 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="{\textstyle \tanh {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \tanh {\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63fa2fc97d762f87535db07dbe22273f2518f05d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.697ex; height:3.509ex;" alt="{\textstyle \tanh {\tfrac {1}{2}}}"></span> in terms of the <a href="/wiki/Exponential_function#Complex_plane" title="Exponential function">exponential</a>, then we can see 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="{\textstyle s,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>s</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle s,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d26efb726dea050d191f29941d5a1f005a3289a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.737ex; height:2.009ex;" alt="{\textstyle s,}"></span> <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 \exp \phi i,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>i</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp \phi i,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e1a7962be704bc88bc0b81d9c22cf29f414cc91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.774ex; height:2.509ex;" alt="{\displaystyle \exp \phi i,}"></span> 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 \exp \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dadd56637a0dd008b658bf676a868949f366078" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.453ex; height:2.509ex;" alt="{\displaystyle \exp \psi }"></span> are all <a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a> of each-other (specifically, rotations of the <a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a>): </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 {\begin{aligned}s&=i{\frac {1-e^{\phi i}}{1+e^{\phi i}}}={\frac {e^{\psi }-1}{e^{\psi }+1}},\\[10mu]i{\frac {s-i}{s+i}}&=\exp \phi i\quad ={\frac {e^{\psi }-i}{e^{\psi }+i}},\\[10mu]{\frac {1+s}{1-s}}&=i{\frac {i+e^{\phi i}}{i-e^{\phi i}}}\,=\exp \psi .\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="0.856em 0.856em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>s</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mi>i</mi> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mo>−</mo> <mi>i</mi> </mrow> <mrow> <mi>s</mi> <mo>+</mo> <mi>i</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>i</mi> <mspace width="1em" /> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msup> <mo>−</mo> <mi>i</mi> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msup> <mo>+</mo> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>s</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>s</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mi>i</mi> </mrow> </msup> </mrow> <mrow> <mi>i</mi> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s&=i{\frac {1-e^{\phi i}}{1+e^{\phi i}}}={\frac {e^{\psi }-1}{e^{\psi }+1}},\\[10mu]i{\frac {s-i}{s+i}}&=\exp \phi i\quad ={\frac {e^{\psi }-i}{e^{\psi }+i}},\\[10mu]{\frac {1+s}{1-s}}&=i{\frac {i+e^{\phi i}}{i-e^{\phi i}}}\,=\exp \psi .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d92cae8f3cf95d4e45c8676a07e2196614fee0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.005ex; width:29.696ex; height:21.176ex;" alt="{\displaystyle {\begin{aligned}s&=i{\frac {1-e^{\phi i}}{1+e^{\phi i}}}={\frac {e^{\psi }-1}{e^{\psi }+1}},\\[10mu]i{\frac {s-i}{s+i}}&=\exp \phi i\quad ={\frac {e^{\psi }-i}{e^{\psi }+i}},\\[10mu]{\frac {1+s}{1-s}}&=i{\frac {i+e^{\phi i}}{i-e^{\phi i}}}\,=\exp \psi .\end{aligned}}}"></span></dd></dl> <p>For real values 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="{\textstyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d738e3571903ec4e786923ddbd817cd147cb5a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\textstyle \psi }"></span> 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="{\textstyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b36dc4887c86d6f01183292a5d2f984e23318c10" 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="{\textstyle \phi }"></span> 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 |\phi |<{\tfrac {1}{2}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi |<{\tfrac {1}{2}}\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f3d338d77072c0d07ad3744fe4c7cc059af8008" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.768ex; height:3.509ex;" alt="{\displaystyle |\phi |<{\tfrac {1}{2}}\pi }"></span>, these Möbius transformations can be written in terms of trigonometric functions in several ways, </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 {\begin{aligned}\exp \psi &=\sec \phi +\tan \phi =\tan {\tfrac {1}{2}}{\bigl (}{\tfrac {1}{2}}\pi +\phi {\bigr )}\\[6mu]&={\frac {1+\tan {\tfrac {1}{2}}\phi }{1-\tan {\tfrac {1}{2}}\phi }}={\sqrt {\frac {1+\sin \phi }{1-\sin \phi }}},\\[12mu]\exp \phi i&=\operatorname {sech} \psi +i\tanh \psi =\tanh {\tfrac {1}{2}}{\bigl (}{-{\tfrac {1}{2}}}\pi i+\psi {\bigr )}\\[6mu]&={\frac {1+i\tanh {\tfrac {1}{2}}\psi }{1-i\tanh {\tfrac {1}{2}}\psi }}={\sqrt {\frac {1+i\sinh \psi }{1-i\sinh \psi }}}.\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="0.633em 0.967em 0.633em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>exp</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sec</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>+</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>exp</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>i</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sech</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>+</mo> <mi>i</mi> <mi>tanh</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>=</mo> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mi>π</mi> <mi>i</mi> <mo>+</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>ψ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>ψ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mi>sinh</mi> <mo>⁡</mo> <mi>ψ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mi>sinh</mi> <mo>⁡</mo> <mi>ψ</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\exp \psi &=\sec \phi +\tan \phi =\tan {\tfrac {1}{2}}{\bigl (}{\tfrac {1}{2}}\pi +\phi {\bigr )}\\[6mu]&={\frac {1+\tan {\tfrac {1}{2}}\phi }{1-\tan {\tfrac {1}{2}}\phi }}={\sqrt {\frac {1+\sin \phi }{1-\sin \phi }}},\\[12mu]\exp \phi i&=\operatorname {sech} \psi +i\tanh \psi =\tanh {\tfrac {1}{2}}{\bigl (}{-{\tfrac {1}{2}}}\pi i+\psi {\bigr )}\\[6mu]&={\frac {1+i\tanh {\tfrac {1}{2}}\psi }{1-i\tanh {\tfrac {1}{2}}\psi }}={\sqrt {\frac {1+i\sinh \psi }{1-i\sinh \psi }}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f95aa3728688e9830bd9119ccae4992b11a4f703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.838ex; width:48.613ex; height:26.843ex;" alt="{\displaystyle {\begin{aligned}\exp \psi &=\sec \phi +\tan \phi =\tan {\tfrac {1}{2}}{\bigl (}{\tfrac {1}{2}}\pi +\phi {\bigr )}\\[6mu]&={\frac {1+\tan {\tfrac {1}{2}}\phi }{1-\tan {\tfrac {1}{2}}\phi }}={\sqrt {\frac {1+\sin \phi }{1-\sin \phi }}},\\[12mu]\exp \phi i&=\operatorname {sech} \psi +i\tanh \psi =\tanh {\tfrac {1}{2}}{\bigl (}{-{\tfrac {1}{2}}}\pi i+\psi {\bigr )}\\[6mu]&={\frac {1+i\tanh {\tfrac {1}{2}}\psi }{1-i\tanh {\tfrac {1}{2}}\psi }}={\sqrt {\frac {1+i\sinh \psi }{1-i\sinh \psi }}}.\end{aligned}}}"></span></dd></dl> <p>These give further expressions 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 \operatorname {gd} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>gd</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gd} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dad73bd4edce09e79f274f0bfcd574c0c32a46a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \operatorname {gd} }"></span> 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 \operatorname {gd} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gd} ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a79e1a6ee5398ea160393c746ff96902f6857fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.788ex; height:3.009ex;" alt="{\displaystyle \operatorname {gd} ^{-1}}"></span> for real arguments 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 |\phi |<{\tfrac {1}{2}}\pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi |<{\tfrac {1}{2}}\pi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad50d61647f0c09b281efef44887ad39c5cdab78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.415ex; height:3.509ex;" alt="{\displaystyle |\phi |<{\tfrac {1}{2}}\pi .}"></span> For example,<sup id="cite_ref-weinstein_8-0" class="reference"><a href="#cite_note-weinstein-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </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 {\begin{aligned}\operatorname {gd} \psi &=2\arctan e^{\psi }-{\tfrac {1}{2}}\pi ,\\[6mu]\operatorname {gd} ^{-1}\phi &=\log(\sec \phi +\tan \phi ).\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="0.633em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>gd</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sec</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {gd} \psi &=2\arctan e^{\psi }-{\tfrac {1}{2}}\pi ,\\[6mu]\operatorname {gd} ^{-1}\phi &=\log(\sec \phi +\tan \phi ).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9899f14b480bfe2561cb64a139a22f14cd0c16d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:28.564ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {gd} \psi &=2\arctan e^{\psi }-{\tfrac {1}{2}}\pi ,\\[6mu]\operatorname {gd} ^{-1}\phi &=\log(\sec \phi +\tan \phi ).\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Complex_values">Complex values</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gudermannian_function&action=edit&section=2" title="Edit section: Complex values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Gudermannian_conformal_map.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Gudermannian_conformal_map.png/330px-Gudermannian_conformal_map.png" decoding="async" width="330" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Gudermannian_conformal_map.png/495px-Gudermannian_conformal_map.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Gudermannian_conformal_map.png/660px-Gudermannian_conformal_map.png 2x" data-file-width="1852" data-file-height="1264" /></a><figcaption>The Gudermannian function <span class="texhtml"><i>z</i> ↦ gd <i>z</i></span> is a conformal map from an infinite strip to an infinite strip. It can be broken into two parts: a map <span class="texhtml"><i>z</i> ↦ tanh <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>z</i></span> from one infinite strip to the complex unit disk and a map <span class="texhtml"><i>ζ</i> ↦ 2 arctan <i>ζ</i></span> from the disk to the other infinite strip.</figcaption></figure> <p>As a <a href="/wiki/Complex_analysis" title="Complex analysis">function of a complex variable</a>, <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="{\textstyle z\mapsto w=\operatorname {gd} z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">↦</mo> <mi>w</mi> <mo>=</mo> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle z\mapsto w=\operatorname {gd} z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bdb4144b7311fd07edc54a378e64335c2ef276e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.395ex; height:2.509ex;" alt="{\textstyle z\mapsto w=\operatorname {gd} z}"></span> <a href="/wiki/Conformal_map" title="Conformal map">conformally maps</a> the infinite strip <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="{\textstyle \left|\operatorname {Im} z\right|\leq {\tfrac {1}{2}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>Im</mi> <mo>⁡</mo> <mi>z</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left|\operatorname {Im} z\right|\leq {\tfrac {1}{2}}\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48fb80f96e3b703ba40d66546b707dee5f51bfc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.633ex; height:3.509ex;" alt="{\textstyle \left|\operatorname {Im} z\right|\leq {\tfrac {1}{2}}\pi }"></span> to the infinite strip <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="{\textstyle \left|\operatorname {Re} w\right|\leq {\tfrac {1}{2}}\pi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>Re</mi> <mo>⁡</mo> <mi>w</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left|\operatorname {Re} w\right|\leq {\tfrac {1}{2}}\pi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8c7d6d15edeecf351d965f0adab59a04fe1c0e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.823ex; height:3.509ex;" alt="{\textstyle \left|\operatorname {Re} w\right|\leq {\tfrac {1}{2}}\pi ,}"></span> while <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="{\textstyle w\mapsto z=\operatorname {gd} ^{-1}w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>w</mi> <mo stretchy="false">↦</mo> <mi>z</mi> <mo>=</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle w\mapsto z=\operatorname {gd} ^{-1}w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d57e4ce6ed6e550c007a694fdfe39ec274c0f49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.304ex; height:3.009ex;" alt="{\textstyle w\mapsto z=\operatorname {gd} ^{-1}w}"></span> conformally maps the infinite strip <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="{\textstyle \left|\operatorname {Re} w\right|\leq {\tfrac {1}{2}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>Re</mi> <mo>⁡</mo> <mi>w</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left|\operatorname {Re} w\right|\leq {\tfrac {1}{2}}\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/552017bf9a52cb9c0ab424ce172c15a6a2b1257c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.176ex; height:3.509ex;" alt="{\textstyle \left|\operatorname {Re} w\right|\leq {\tfrac {1}{2}}\pi }"></span> to the infinite strip <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="{\textstyle \left|\operatorname {Im} z\right|\leq {\tfrac {1}{2}}\pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>Im</mi> <mo>⁡</mo> <mi>z</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left|\operatorname {Im} z\right|\leq {\tfrac {1}{2}}\pi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4b53669df6286ddb122a107f08b815eed91770c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.28ex; height:3.509ex;" alt="{\textstyle \left|\operatorname {Im} z\right|\leq {\tfrac {1}{2}}\pi .}"></span> </p><p><a href="/wiki/Analytic_continuation" title="Analytic continuation">Analytically continued</a> by <a href="/wiki/Schwarz_reflection_principle" title="Schwarz reflection principle">reflections</a> to the whole complex plane, <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="{\textstyle z\mapsto w=\operatorname {gd} z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">↦</mo> <mi>w</mi> <mo>=</mo> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle z\mapsto w=\operatorname {gd} z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bdb4144b7311fd07edc54a378e64335c2ef276e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.395ex; height:2.509ex;" alt="{\textstyle z\mapsto w=\operatorname {gd} z}"></span> is a periodic function of period <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="{\textstyle 2\pi i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2\pi i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92f94a26afea8d967ee55013aca00dc6c9ab3e77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.297ex; height:2.176ex;" alt="{\textstyle 2\pi i}"></span> which sends any infinite strip of "height" <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="{\textstyle 2\pi i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2\pi i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92f94a26afea8d967ee55013aca00dc6c9ab3e77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.297ex; height:2.176ex;" alt="{\textstyle 2\pi i}"></span> onto the strip <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="{\textstyle -\pi <\operatorname {Re} w\leq \pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mi>π</mi> <mo><</mo> <mi>Re</mi> <mo>⁡</mo> <mi>w</mi> <mo>≤</mo> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -\pi <\operatorname {Re} w\leq \pi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a4cd438bb7cdaf081d16d7d8b485dfe4b81cbb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.11ex; height:2.343ex;" alt="{\textstyle -\pi <\operatorname {Re} w\leq \pi .}"></span> Likewise, extended to the whole complex plane, <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="{\textstyle w\mapsto z=\operatorname {gd} ^{-1}w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>w</mi> <mo stretchy="false">↦</mo> <mi>z</mi> <mo>=</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle w\mapsto z=\operatorname {gd} ^{-1}w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d57e4ce6ed6e550c007a694fdfe39ec274c0f49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.304ex; height:3.009ex;" alt="{\textstyle w\mapsto z=\operatorname {gd} ^{-1}w}"></span> is a periodic function of period <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="{\textstyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/584b74e57567454e6a48e040ec231214be5b8c54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\textstyle 2\pi }"></span> which sends any infinite strip of "width" <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="{\textstyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/584b74e57567454e6a48e040ec231214be5b8c54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\textstyle 2\pi }"></span> onto the strip <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="{\textstyle -\pi <\operatorname {Im} z\leq \pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mi>π</mi> <mo><</mo> <mi>Im</mi> <mo>⁡</mo> <mi>z</mi> <mo>≤</mo> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -\pi <\operatorname {Im} z\leq \pi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8436da86329e92801451943c901a52bead35eea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.567ex; height:2.343ex;" alt="{\textstyle -\pi <\operatorname {Im} z\leq \pi .}"></span><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> For all points in the complex plane, these functions can be correctly written as: </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 {\begin{aligned}\operatorname {gd} z&={2\arctan }{\bigl (}\tanh {\tfrac {1}{2}}z\,{\bigr )},\\[5mu]\operatorname {gd} ^{-1}w&={2\operatorname {artanh} }{\bigl (}\tan {\tfrac {1}{2}}w\,{\bigr )}.\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="0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>arctan</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>z</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>w</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>artanh</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>w</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {gd} z&={2\arctan }{\bigl (}\tanh {\tfrac {1}{2}}z\,{\bigr )},\\[5mu]\operatorname {gd} ^{-1}w&={2\operatorname {artanh} }{\bigl (}\tan {\tfrac {1}{2}}w\,{\bigr )}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23d2333122dc1bc39541bfcaa136cacd8b8c46a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:30.041ex; height:8.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {gd} z&={2\arctan }{\bigl (}\tanh {\tfrac {1}{2}}z\,{\bigr )},\\[5mu]\operatorname {gd} ^{-1}w&={2\operatorname {artanh} }{\bigl (}\tan {\tfrac {1}{2}}w\,{\bigr )}.\end{aligned}}}"></span></dd></dl> <p>For the <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="{\textstyle \operatorname {gd} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>gd</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {gd} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e8496c6ca1328821c3f42dd1173a63d6dacbd05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\textstyle \operatorname {gd} }"></span> 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="{\textstyle \operatorname {gd} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {gd} ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c67c70418c886a7bc07a9078fecbef157b393a99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.788ex; height:3.009ex;" alt="{\textstyle \operatorname {gd} ^{-1}}"></span> functions to remain invertible with these extended domains, we might consider each to be a <a href="/wiki/Multivalued_function" title="Multivalued function">multivalued function</a> (perhaps <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="{\textstyle \operatorname {Gd} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>Gd</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {Gd} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa53c8a5b4ddec79a30cd82af850b08a2de6e190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.117ex; height:2.176ex;" alt="{\textstyle \operatorname {Gd} }"></span> 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="{\textstyle \operatorname {Gd} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>Gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {Gd} ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fa17b13c05f58bdf75cb68ba29430bb1bf2012e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.45ex; height:2.676ex;" alt="{\textstyle \operatorname {Gd} ^{-1}}"></span>, 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="{\textstyle \operatorname {gd} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>gd</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {gd} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e8496c6ca1328821c3f42dd1173a63d6dacbd05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\textstyle \operatorname {gd} }"></span> 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="{\textstyle \operatorname {gd} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {gd} ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c67c70418c886a7bc07a9078fecbef157b393a99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.788ex; height:3.009ex;" alt="{\textstyle \operatorname {gd} ^{-1}}"></span> the <a href="/wiki/Principal_branch" title="Principal branch">principal branch</a>) or consider their domains and codomains as <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a>. </p><p>If <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="{\textstyle u+iv=\operatorname {gd} (x+iy),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>u</mi> <mo>+</mo> <mi>i</mi> <mi>v</mi> <mo>=</mo> <mi>gd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle u+iv=\operatorname {gd} (x+iy),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31664a4c0b6bb4e267ab655eb14b635083c3981b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.238ex; height:2.843ex;" alt="{\textstyle u+iv=\operatorname {gd} (x+iy),}"></span> then the real and imaginary components <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="{\textstyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24e12e26e505ed5b02c7648a89bbc6737038b2de" 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="{\textstyle u}"></span> 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="{\textstyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e970b082b4c08f1583d01d713815a0900fab6a8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\textstyle v}"></span> can be found by:<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </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 \tan u={\frac {\sinh x}{\cos y}},\quad \tanh v={\frac {\sin y}{\cosh x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sinh</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>tanh</mi> <mo>⁡</mo> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>y</mi> </mrow> <mrow> <mi>cosh</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan u={\frac {\sinh x}{\cos y}},\quad \tanh v={\frac {\sin y}{\cosh x}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e4e80ae5f98b4aca39d6f4645d456efbc8b75d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.101ex; height:5.843ex;" alt="{\displaystyle \tan u={\frac {\sinh x}{\cos y}},\quad \tanh v={\frac {\sin y}{\cosh x}}.}"></span></dd></dl> <p>(In practical implementation, make sure to use the <a href="/wiki/Atan2" title="Atan2">2-argument arctangent</a>, <span class="nowrap"><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="{\textstyle u=\operatorname {atan2} (\sinh x,\cos y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mi>atan2</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sinh</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> <mi>cos</mi> <mo>⁡</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle u=\operatorname {atan2} (\sinh x,\cos y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6343defd6ee31e2654a7660a846879f01f5e262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.475ex; height:2.843ex;" alt="{\textstyle u=\operatorname {atan2} (\sinh x,\cos y)}"></span>.)</span> </p><p>Likewise, if <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="{\textstyle x+iy=\operatorname {gd} ^{-1}(u+iv),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo>=</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <mi>i</mi> <mi>v</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x+iy=\operatorname {gd} ^{-1}(u+iv),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7f119c52abd01acdc70fab29445e38efaece085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.571ex; height:3.176ex;" alt="{\textstyle x+iy=\operatorname {gd} ^{-1}(u+iv),}"></span> then components <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="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" 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="{\textstyle x}"></span> 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="{\textstyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db9936ddb2761b76fa640fb275cb5d1fa4d6fa23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\textstyle y}"></span> can be found by:<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </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 \tanh x={\frac {\sin u}{\cosh v}},\quad \tan y={\frac {\sinh v}{\cos u}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tanh</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>u</mi> </mrow> <mrow> <mi>cosh</mi> <mo>⁡</mo> <mi>v</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>tan</mi> <mo>⁡</mo> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sinh</mi> <mo>⁡</mo> <mi>v</mi> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>u</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tanh x={\frac {\sin u}{\cosh v}},\quad \tan y={\frac {\sinh v}{\cos u}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95bfbf84aab10fd2dbba5b857a4f916a3e2bea64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.725ex; height:5.509ex;" alt="{\displaystyle \tanh x={\frac {\sin u}{\cosh v}},\quad \tan y={\frac {\sinh v}{\cos u}}.}"></span></dd></dl> <p>Multiplying these together reveals the additional identity<sup id="cite_ref-weinstein_8-1" class="reference"><a href="#cite_note-weinstein-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </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 \tanh x\,\tan y=\tan u\,\tanh v.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tanh</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mi>tan</mi> <mo>⁡</mo> <mi>y</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mi>u</mi> <mspace width="thinmathspace" /> <mi>tanh</mi> <mo>⁡</mo> <mi>v</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tanh x\,\tan y=\tan u\,\tanh v.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c30395d21f39b27da97ab2bfad9e860a51cb56fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.808ex; height:2.509ex;" alt="{\displaystyle \tanh x\,\tan y=\tan u\,\tanh v.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Symmetries">Symmetries</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gudermannian_function&action=edit&section=3" title="Edit section: Symmetries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The two functions can be thought of as rotations or reflections of each-other, with a similar relationship as <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="{\textstyle \sinh iz=i\sin z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>sinh</mi> <mo>⁡</mo> <mi>i</mi> <mi>z</mi> <mo>=</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sinh iz=i\sin z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9553444ba57f192fd1fa063907a49d125ff7aec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.045ex; height:2.176ex;" alt="{\textstyle \sinh iz=i\sin z}"></span> <a href="/wiki/Hyperbolic_functions#Hyperbolic_functions_for_complex_numbers" title="Hyperbolic functions">between sine and hyperbolic sine</a>:<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </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 {\begin{aligned}\operatorname {gd} iz&=i\operatorname {gd} ^{-1}z,\\[5mu]\operatorname {gd} ^{-1}iz&=i\operatorname {gd} z.\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="0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>gd</mi> <mo>⁡</mo> <mi>i</mi> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>i</mi> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>z</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>i</mi> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>i</mi> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {gd} iz&=i\operatorname {gd} ^{-1}z,\\[5mu]\operatorname {gd} ^{-1}iz&=i\operatorname {gd} z.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32f3f162617f8e2fbf080a44980b28db1dac579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.015ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {gd} iz&=i\operatorname {gd} ^{-1}z,\\[5mu]\operatorname {gd} ^{-1}iz&=i\operatorname {gd} z.\end{aligned}}}"></span></dd></dl> <p>The functions are both <a href="/wiki/Even_and_odd_functions" title="Even and odd functions">odd</a> and they commute with <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugation</a>. That is, a reflection across the real or imaginary axis in the domain results in the same reflection in the codomain: </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 {\begin{aligned}\operatorname {gd} (-z)&=-\operatorname {gd} z,&\quad \operatorname {gd} {\bar {z}}&={\overline {\operatorname {gd} z}},&\quad \operatorname {gd} (-{\bar {z}})&=-{\overline {\operatorname {gd} z}},\\[5mu]\operatorname {gd} ^{-1}(-z)&=-\operatorname {gd} ^{-1}z,&\quad \operatorname {gd} ^{-1}{\bar {z}}&={\overline {\operatorname {gd} ^{-1}z}},&\quad \operatorname {gd} ^{-1}(-{\bar {z}})&=-{\overline {\operatorname {gd} ^{-1}z}}.\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="0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>gd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <mi>gd</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <mi>gd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>z</mi> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>z</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>z</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {gd} (-z)&=-\operatorname {gd} z,&\quad \operatorname {gd} {\bar {z}}&={\overline {\operatorname {gd} z}},&\quad \operatorname {gd} (-{\bar {z}})&=-{\overline {\operatorname {gd} z}},\\[5mu]\operatorname {gd} ^{-1}(-z)&=-\operatorname {gd} ^{-1}z,&\quad \operatorname {gd} ^{-1}{\bar {z}}&={\overline {\operatorname {gd} ^{-1}z}},&\quad \operatorname {gd} ^{-1}(-{\bar {z}})&=-{\overline {\operatorname {gd} ^{-1}z}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180914d120718119ca1f5bce88c563adca935cbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.296ex; margin-bottom: -0.208ex; width:74.62ex; height:8.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {gd} (-z)&=-\operatorname {gd} z,&\quad \operatorname {gd} {\bar {z}}&={\overline {\operatorname {gd} z}},&\quad \operatorname {gd} (-{\bar {z}})&=-{\overline {\operatorname {gd} z}},\\[5mu]\operatorname {gd} ^{-1}(-z)&=-\operatorname {gd} ^{-1}z,&\quad \operatorname {gd} ^{-1}{\bar {z}}&={\overline {\operatorname {gd} ^{-1}z}},&\quad \operatorname {gd} ^{-1}(-{\bar {z}})&=-{\overline {\operatorname {gd} ^{-1}z}}.\end{aligned}}}"></span></dd></dl> <p>The functions are <a href="/wiki/Periodic_function" title="Periodic function">periodic</a>, with periods <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="{\textstyle 2\pi i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2\pi i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92f94a26afea8d967ee55013aca00dc6c9ab3e77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.297ex; height:2.176ex;" alt="{\textstyle 2\pi i}"></span> 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="{\textstyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/584b74e57567454e6a48e040ec231214be5b8c54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\textstyle 2\pi }"></span>: </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 {\begin{aligned}\operatorname {gd} (z+2\pi i)&=\operatorname {gd} z,\\[5mu]\operatorname {gd} ^{-1}(z+2\pi )&=\operatorname {gd} ^{-1}z.\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="0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>gd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>z</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {gd} (z+2\pi i)&=\operatorname {gd} z,\\[5mu]\operatorname {gd} ^{-1}(z+2\pi )&=\operatorname {gd} ^{-1}z.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dad8e99fb169c5a17b03343210d21d41c7f19c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:23.78ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {gd} (z+2\pi i)&=\operatorname {gd} z,\\[5mu]\operatorname {gd} ^{-1}(z+2\pi )&=\operatorname {gd} ^{-1}z.\end{aligned}}}"></span></dd></dl> <p>A translation in the domain 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="{\textstyle \operatorname {gd} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>gd</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {gd} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e8496c6ca1328821c3f42dd1173a63d6dacbd05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\textstyle \operatorname {gd} }"></span> by <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="{\textstyle \pm \pi i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>±</mo> <mi>π</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \pm \pi i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6446f42e6aac3d44906105503f4b762c358665be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.943ex; height:2.176ex;" alt="{\textstyle \pm \pi i}"></span> results in a half-turn rotation and translation in the codomain by one 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="{\textstyle \pm \pi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>±</mo> <mi>π</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \pm \pi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dae1d4614b7b9d8367d047403284237b395d1e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.787ex; height:2.509ex;" alt="{\textstyle \pm \pi ,}"></span> and vice versa 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="{\textstyle \operatorname {gd} ^{-1}\colon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {gd} ^{-1}\colon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b88a9e6ea28b014d77421776ea5148ca97eec6c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.435ex; height:3.009ex;" alt="{\textstyle \operatorname {gd} ^{-1}\colon }"></span><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </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 {\begin{aligned}\operatorname {gd} ({\pm \pi i}+z)&={\begin{cases}\pi -\operatorname {gd} z\quad &{\mbox{if }}\ \ \operatorname {Re} z\geq 0,\\[5mu]-\pi -\operatorname {gd} z\quad &{\mbox{if }}\ \ \operatorname {Re} z<0,\end{cases}}\\[15mu]\operatorname {gd} ^{-1}({\pm \pi }+z)&={\begin{cases}\pi i-\operatorname {gd} ^{-1}z\quad &{\mbox{if }}\ \ \operatorname {Im} z\geq 0,\\[3mu]-\pi i-\operatorname {gd} ^{-1}z\quad &{\mbox{if }}\ \ \operatorname {Im} z<0.\end{cases}}\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="1.133em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>gd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mi>π</mi> <mi>i</mi> </mrow> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.478em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>π</mi> <mo>−</mo> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="1em" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mtext> </mtext> <mtext> </mtext> <mi>Re</mi> <mo>⁡</mo> <mi>z</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>π</mi> <mo>−</mo> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="1em" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mtext> </mtext> <mtext> </mtext> <mi>Re</mi> <mo>⁡</mo> <mi>z</mi> <mo><</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mi>π</mi> </mrow> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.367em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>π</mi> <mi>i</mi> <mo>−</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>z</mi> <mspace width="1em" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mtext> </mtext> <mtext> </mtext> <mi>Im</mi> <mo>⁡</mo> <mi>z</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>π</mi> <mi>i</mi> <mo>−</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>z</mi> <mspace width="1em" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mtext> </mtext> <mtext> </mtext> <mi>Im</mi> <mo>⁡</mo> <mi>z</mi> <mo><</mo> <mn>0.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {gd} ({\pm \pi i}+z)&={\begin{cases}\pi -\operatorname {gd} z\quad &{\mbox{if }}\ \ \operatorname {Re} z\geq 0,\\[5mu]-\pi -\operatorname {gd} z\quad &{\mbox{if }}\ \ \operatorname {Re} z<0,\end{cases}}\\[15mu]\operatorname {gd} ^{-1}({\pm \pi }+z)&={\begin{cases}\pi i-\operatorname {gd} ^{-1}z\quad &{\mbox{if }}\ \ \operatorname {Im} z\geq 0,\\[3mu]-\pi i-\operatorname {gd} ^{-1}z\quad &{\mbox{if }}\ \ \operatorname {Im} z<0.\end{cases}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71d910f6f58789c6e15efc9e96c171b8e56ce821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:50.348ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {gd} ({\pm \pi i}+z)&={\begin{cases}\pi -\operatorname {gd} z\quad &{\mbox{if }}\ \ \operatorname {Re} z\geq 0,\\[5mu]-\pi -\operatorname {gd} z\quad &{\mbox{if }}\ \ \operatorname {Re} z<0,\end{cases}}\\[15mu]\operatorname {gd} ^{-1}({\pm \pi }+z)&={\begin{cases}\pi i-\operatorname {gd} ^{-1}z\quad &{\mbox{if }}\ \ \operatorname {Im} z\geq 0,\\[3mu]-\pi i-\operatorname {gd} ^{-1}z\quad &{\mbox{if }}\ \ \operatorname {Im} z<0.\end{cases}}\end{aligned}}}"></span></dd></dl> <p>A reflection in the domain 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="{\textstyle \operatorname {gd} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>gd</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {gd} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e8496c6ca1328821c3f42dd1173a63d6dacbd05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\textstyle \operatorname {gd} }"></span> across either of the lines <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="{\textstyle x\pm {\tfrac {1}{2}}\pi i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x\pm {\tfrac {1}{2}}\pi i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc59dfc13d3c5f8898b09234dce267074d4f3ab8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.963ex; height:3.509ex;" alt="{\textstyle x\pm {\tfrac {1}{2}}\pi i}"></span> results in a reflection in the codomain across one of the lines <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="{\textstyle \pm {\tfrac {1}{2}}\pi +yi,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>+</mo> <mi>y</mi> <mi>i</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \pm {\tfrac {1}{2}}\pi +yi,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e92a89b566ab7cb2725ca216e17d7f3241a6153f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.243ex; height:3.509ex;" alt="{\textstyle \pm {\tfrac {1}{2}}\pi +yi,}"></span> and vice versa 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="{\textstyle \operatorname {gd} ^{-1}\colon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {gd} ^{-1}\colon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b88a9e6ea28b014d77421776ea5148ca97eec6c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.435ex; height:3.009ex;" alt="{\textstyle \operatorname {gd} ^{-1}\colon }"></span> </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 {\begin{aligned}\operatorname {gd} ({\pm \pi i}+{\bar {z}})&={\begin{cases}\pi -{\overline {\operatorname {gd} z}}\quad &{\mbox{if }}\ \ \operatorname {Re} z\geq 0,\\[5mu]-\pi -{\overline {\operatorname {gd} z}}\quad &{\mbox{if }}\ \ \operatorname {Re} z<0,\end{cases}}\\[15mu]\operatorname {gd} ^{-1}({\pm \pi }-{\bar {z}})&={\begin{cases}\pi i+{\overline {\operatorname {gd} ^{-1}z}}\quad &{\mbox{if }}\ \ \operatorname {Im} z\geq 0,\\[3mu]-\pi i+{\overline {\operatorname {gd} ^{-1}z}}\quad &{\mbox{if }}\ \ \operatorname {Im} z<0.\end{cases}}\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="1.133em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>gd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mi>π</mi> <mi>i</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.478em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>π</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="1em" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mtext> </mtext> <mtext> </mtext> <mi>Re</mi> <mo>⁡</mo> <mi>z</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>π</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="1em" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mtext> </mtext> <mtext> </mtext> <mi>Re</mi> <mo>⁡</mo> <mi>z</mi> <mo><</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mi>π</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.367em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>π</mi> <mi>i</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>z</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="1em" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mtext> </mtext> <mtext> </mtext> <mi>Im</mi> <mo>⁡</mo> <mi>z</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>π</mi> <mi>i</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>z</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="1em" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mtext> </mtext> <mtext> </mtext> <mi>Im</mi> <mo>⁡</mo> <mi>z</mi> <mo><</mo> <mn>0.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {gd} ({\pm \pi i}+{\bar {z}})&={\begin{cases}\pi -{\overline {\operatorname {gd} z}}\quad &{\mbox{if }}\ \ \operatorname {Re} z\geq 0,\\[5mu]-\pi -{\overline {\operatorname {gd} z}}\quad &{\mbox{if }}\ \ \operatorname {Re} z<0,\end{cases}}\\[15mu]\operatorname {gd} ^{-1}({\pm \pi }-{\bar {z}})&={\begin{cases}\pi i+{\overline {\operatorname {gd} ^{-1}z}}\quad &{\mbox{if }}\ \ \operatorname {Im} z\geq 0,\\[3mu]-\pi i+{\overline {\operatorname {gd} ^{-1}z}}\quad &{\mbox{if }}\ \ \operatorname {Im} z<0.\end{cases}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/618b9377241278124edfde77b44a4582fdfab5d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:50.805ex; height:17.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {gd} ({\pm \pi i}+{\bar {z}})&={\begin{cases}\pi -{\overline {\operatorname {gd} z}}\quad &{\mbox{if }}\ \ \operatorname {Re} z\geq 0,\\[5mu]-\pi -{\overline {\operatorname {gd} z}}\quad &{\mbox{if }}\ \ \operatorname {Re} z<0,\end{cases}}\\[15mu]\operatorname {gd} ^{-1}({\pm \pi }-{\bar {z}})&={\begin{cases}\pi i+{\overline {\operatorname {gd} ^{-1}z}}\quad &{\mbox{if }}\ \ \operatorname {Im} z\geq 0,\\[3mu]-\pi i+{\overline {\operatorname {gd} ^{-1}z}}\quad &{\mbox{if }}\ \ \operatorname {Im} z<0.\end{cases}}\end{aligned}}}"></span></dd></dl> <p>This is related to the identity </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 \tanh {\tfrac {1}{2}}({\pi i}\pm z)=\tan {\tfrac {1}{2}}({\pi }\mp \operatorname {gd} z).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mi>i</mi> </mrow> <mo>±</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> <mo>∓</mo> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tanh {\tfrac {1}{2}}({\pi i}\pm z)=\tan {\tfrac {1}{2}}({\pi }\mp \operatorname {gd} z).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8305fde49e12cd93b5f66d327559b45e3cff764" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:33.632ex; height:3.509ex;" alt="{\displaystyle \tanh {\tfrac {1}{2}}({\pi i}\pm z)=\tan {\tfrac {1}{2}}({\pi }\mp \operatorname {gd} z).}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Specific_values">Specific values</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gudermannian_function&action=edit&section=4" title="Edit section: Specific values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A few specific values (where <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="{\textstyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7672961cb69498135a93484fe61fedd72996ad03" 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="{\textstyle \infty }"></span> indicates the limit at one end of the infinite strip):<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </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 {\begin{aligned}\operatorname {gd} (0)&=0,&\quad {\operatorname {gd} }{\bigl (}{\pm {\log }{\bigl (}2+{\sqrt {3}}{\bigr )}}{\bigr )}&=\pm {\tfrac {1}{3}}\pi ,\\[5mu]\operatorname {gd} (\pi i)&=\pi ,&\quad {\operatorname {gd} }{\bigl (}{\pm {\tfrac {1}{3}}}\pi i{\bigr )}&=\pm {\log }{\bigl (}2+{\sqrt {3}}{\bigr )}i,\\[5mu]\operatorname {gd} ({\pm \infty })&=\pm {\tfrac {1}{2}}\pi ,&\quad {\operatorname {gd} }{\bigl (}{\pm {\log }{\bigl (}1+{\sqrt {2}}{\bigr )}}{\bigr )}&=\pm {\tfrac {1}{4}}\pi ,\\[5mu]{\operatorname {gd} }{\bigl (}{\pm {\tfrac {1}{2}}}\pi i{\bigr )}&=\pm \infty i,&\quad {\operatorname {gd} }{\bigl (}{\pm {\tfrac {1}{4}}}\pi i{\bigr )}&=\pm {\log }{\bigl (}1+{\sqrt {2}}{\bigr )}i,\\[5mu]&&{\operatorname {gd} }{\bigl (}{\log }{\bigl (}1+{\sqrt {2}}{\bigr )}\pm {\tfrac {1}{2}}\pi i{\bigr )}&={\tfrac {1}{2}}\pi \pm {\log }{\bigl (}1+{\sqrt {2}}{\bigr )}i,\\[5mu]&&{\operatorname {gd} }{\bigl (}{-\log }{\bigl (}1+{\sqrt {2}}{\bigr )}\pm {\tfrac {1}{2}}\pi i{\bigr )}&=-{\tfrac {1}{2}}\pi \pm {\log }{\bigl (}1+{\sqrt {2}}{\bigr )}i.\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="0.578em 0.578em 0.578em 0.578em 0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>gd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi>gd</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>log</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>gd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>i</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>π</mi> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi>gd</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mrow> <mi>π</mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>log</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mi>i</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>gd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mi mathvariant="normal">∞</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi>gd</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>log</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>gd</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mi>π</mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mi mathvariant="normal">∞</mi> <mi>i</mi> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi>gd</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mrow> <mi>π</mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>log</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mi>i</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>gd</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>log</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>log</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mi>i</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>gd</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>log</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>log</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mi>i</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {gd} (0)&=0,&\quad {\operatorname {gd} }{\bigl (}{\pm {\log }{\bigl (}2+{\sqrt {3}}{\bigr )}}{\bigr )}&=\pm {\tfrac {1}{3}}\pi ,\\[5mu]\operatorname {gd} (\pi i)&=\pi ,&\quad {\operatorname {gd} }{\bigl (}{\pm {\tfrac {1}{3}}}\pi i{\bigr )}&=\pm {\log }{\bigl (}2+{\sqrt {3}}{\bigr )}i,\\[5mu]\operatorname {gd} ({\pm \infty })&=\pm {\tfrac {1}{2}}\pi ,&\quad {\operatorname {gd} }{\bigl (}{\pm {\log }{\bigl (}1+{\sqrt {2}}{\bigr )}}{\bigr )}&=\pm {\tfrac {1}{4}}\pi ,\\[5mu]{\operatorname {gd} }{\bigl (}{\pm {\tfrac {1}{2}}}\pi i{\bigr )}&=\pm \infty i,&\quad {\operatorname {gd} }{\bigl (}{\pm {\tfrac {1}{4}}}\pi i{\bigr )}&=\pm {\log }{\bigl (}1+{\sqrt {2}}{\bigr )}i,\\[5mu]&&{\operatorname {gd} }{\bigl (}{\log }{\bigl (}1+{\sqrt {2}}{\bigr )}\pm {\tfrac {1}{2}}\pi i{\bigr )}&={\tfrac {1}{2}}\pi \pm {\log }{\bigl (}1+{\sqrt {2}}{\bigr )}i,\\[5mu]&&{\operatorname {gd} }{\bigl (}{-\log }{\bigl (}1+{\sqrt {2}}{\bigr )}\pm {\tfrac {1}{2}}\pi i{\bigr )}&=-{\tfrac {1}{2}}\pi \pm {\log }{\bigl (}1+{\sqrt {2}}{\bigr )}i.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62137050459c2dfe94c8ad4890d52bac8dceb818" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.122ex; margin-bottom: -0.216ex; width:74.267ex; height:25.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {gd} (0)&=0,&\quad {\operatorname {gd} }{\bigl (}{\pm {\log }{\bigl (}2+{\sqrt {3}}{\bigr )}}{\bigr )}&=\pm {\tfrac {1}{3}}\pi ,\\[5mu]\operatorname {gd} (\pi i)&=\pi ,&\quad {\operatorname {gd} }{\bigl (}{\pm {\tfrac {1}{3}}}\pi i{\bigr )}&=\pm {\log }{\bigl (}2+{\sqrt {3}}{\bigr )}i,\\[5mu]\operatorname {gd} ({\pm \infty })&=\pm {\tfrac {1}{2}}\pi ,&\quad {\operatorname {gd} }{\bigl (}{\pm {\log }{\bigl (}1+{\sqrt {2}}{\bigr )}}{\bigr )}&=\pm {\tfrac {1}{4}}\pi ,\\[5mu]{\operatorname {gd} }{\bigl (}{\pm {\tfrac {1}{2}}}\pi i{\bigr )}&=\pm \infty i,&\quad {\operatorname {gd} }{\bigl (}{\pm {\tfrac {1}{4}}}\pi i{\bigr )}&=\pm {\log }{\bigl (}1+{\sqrt {2}}{\bigr )}i,\\[5mu]&&{\operatorname {gd} }{\bigl (}{\log }{\bigl (}1+{\sqrt {2}}{\bigr )}\pm {\tfrac {1}{2}}\pi i{\bigr )}&={\tfrac {1}{2}}\pi \pm {\log }{\bigl (}1+{\sqrt {2}}{\bigr )}i,\\[5mu]&&{\operatorname {gd} }{\bigl (}{-\log }{\bigl (}1+{\sqrt {2}}{\bigr )}\pm {\tfrac {1}{2}}\pi i{\bigr )}&=-{\tfrac {1}{2}}\pi \pm {\log }{\bigl (}1+{\sqrt {2}}{\bigr )}i.\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Derivatives">Derivatives</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gudermannian_function&action=edit&section=5" title="Edit section: Derivatives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As the Gudermannian and inverse Gudermannian functions can be defined as the antiderivatives of the hyperbolic secant and circular secant functions, respectively, their derivatives are those secant functions: </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 {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {gd} z&=\operatorname {sech} z,\\[10mu]{\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {gd} ^{-1}z&=\sec z.\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="0.856em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> </mrow> </mfrac> </mrow> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sech</mi> <mo>⁡</mo> <mi>z</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> </mrow> </mfrac> </mrow> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sec</mi> <mo>⁡</mo> <mi>z</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {gd} z&=\operatorname {sech} z,\\[10mu]{\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {gd} ^{-1}z&=\sec z.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5883cae1281f8cb0b8c4fb7ef64310e84ea07de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:20.112ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {gd} z&=\operatorname {sech} z,\\[10mu]{\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {gd} ^{-1}z&=\sec z.\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Argument-addition_identities">Argument-addition identities</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gudermannian_function&action=edit&section=6" title="Edit section: Argument-addition identities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By combining <a href="/wiki/Hyperbolic_functions#Sums_of_arguments" title="Hyperbolic functions">hyperbolic</a> and <a href="/wiki/Trigonometric_functions#Sum_and_difference_formulas" title="Trigonometric functions">circular</a> argument-addition identities, </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 {\begin{aligned}\tanh(z+w)&={\frac {\tanh z+\tanh w}{1+\tanh z\,\tanh w}},\\[10mu]\tan(z+w)&={\frac {\tan z+\tan w}{1-\tan z\,\tan w}},\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="0.856em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>tanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tanh</mi> <mo>⁡</mo> <mi>z</mi> <mo>+</mo> <mi>tanh</mi> <mo>⁡</mo> <mi>w</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>tanh</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="thinmathspace" /> <mi>tanh</mi> <mo>⁡</mo> <mi>w</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>z</mi> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>w</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>z</mi> <mspace width="thinmathspace" /> <mi>tan</mi> <mo>⁡</mo> <mi>w</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tanh(z+w)&={\frac {\tanh z+\tanh w}{1+\tanh z\,\tanh w}},\\[10mu]\tan(z+w)&={\frac {\tan z+\tan w}{1-\tan z\,\tan w}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/105177a58b358faed394dbaeb18c08c6b38f6876" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:34.995ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}\tanh(z+w)&={\frac {\tanh z+\tanh w}{1+\tanh z\,\tanh w}},\\[10mu]\tan(z+w)&={\frac {\tan z+\tan w}{1-\tan z\,\tan w}},\end{aligned}}}"></span></dd></dl> <p>with the <a href="#Circular–hyperbolic_identities">circular–hyperbolic identity</a>, </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 \tan {\tfrac {1}{2}}(\operatorname {gd} z)=\tanh {\tfrac {1}{2}}z,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>z</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan {\tfrac {1}{2}}(\operatorname {gd} z)=\tanh {\tfrac {1}{2}}z,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f165495d4349adf7b2cf89e469cc6669599b81a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.675ex; height:3.509ex;" alt="{\displaystyle \tan {\tfrac {1}{2}}(\operatorname {gd} z)=\tanh {\tfrac {1}{2}}z,}"></span></dd></dl> <p>we have the Gudermannian argument-addition identities: </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 {\begin{aligned}\operatorname {gd} (z+w)&=2\arctan {\frac {\tan {\tfrac {1}{2}}(\operatorname {gd} z)+\tan {\tfrac {1}{2}}(\operatorname {gd} w)}{1+\tan {\tfrac {1}{2}}(\operatorname {gd} z)\,\tan {\tfrac {1}{2}}(\operatorname {gd} w)}},\\[12mu]\operatorname {gd} ^{-1}(z+w)&=2\operatorname {artanh} {\frac {\tanh {\tfrac {1}{2}}(\operatorname {gd} ^{-1}z)+\tanh {\tfrac {1}{2}}(\operatorname {gd} ^{-1}w)}{1-\tanh {\tfrac {1}{2}}(\operatorname {gd} ^{-1}z)\,\tanh {\tfrac {1}{2}}(\operatorname {gd} ^{-1}w)}}.\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="0.967em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>gd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>gd</mi> <mo>⁡</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>gd</mi> <mo>⁡</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>artanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {gd} (z+w)&=2\arctan {\frac {\tan {\tfrac {1}{2}}(\operatorname {gd} z)+\tan {\tfrac {1}{2}}(\operatorname {gd} w)}{1+\tan {\tfrac {1}{2}}(\operatorname {gd} z)\,\tan {\tfrac {1}{2}}(\operatorname {gd} w)}},\\[12mu]\operatorname {gd} ^{-1}(z+w)&=2\operatorname {artanh} {\frac {\tanh {\tfrac {1}{2}}(\operatorname {gd} ^{-1}z)+\tanh {\tfrac {1}{2}}(\operatorname {gd} ^{-1}w)}{1-\tanh {\tfrac {1}{2}}(\operatorname {gd} ^{-1}z)\,\tanh {\tfrac {1}{2}}(\operatorname {gd} ^{-1}w)}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1c23629943ec1c0054a3d0f36066bedc0ee8cba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:61.078ex; height:17.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {gd} (z+w)&=2\arctan {\frac {\tan {\tfrac {1}{2}}(\operatorname {gd} z)+\tan {\tfrac {1}{2}}(\operatorname {gd} w)}{1+\tan {\tfrac {1}{2}}(\operatorname {gd} z)\,\tan {\tfrac {1}{2}}(\operatorname {gd} w)}},\\[12mu]\operatorname {gd} ^{-1}(z+w)&=2\operatorname {artanh} {\frac {\tanh {\tfrac {1}{2}}(\operatorname {gd} ^{-1}z)+\tanh {\tfrac {1}{2}}(\operatorname {gd} ^{-1}w)}{1-\tanh {\tfrac {1}{2}}(\operatorname {gd} ^{-1}z)\,\tanh {\tfrac {1}{2}}(\operatorname {gd} ^{-1}w)}}.\end{aligned}}}"></span></dd></dl> <p>Further argument-addition identities can be written in terms of other circular functions,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> but they require greater care in choosing branches in inverse functions. Notably, </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 {\begin{aligned}\operatorname {gd} (z+w)&=u+v,\quad {\text{where}}\ \tan u={\frac {\sinh z}{\cosh w}},\ \tan v={\frac {\sinh w}{\cosh z}},\\[10mu]\operatorname {gd} ^{-1}(z+w)&=u+v,\quad {\text{where}}\ \tanh u={\frac {\sin z}{\cos w}},\ \tanh v={\frac {\sin w}{\cos z}},\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="0.856em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>gd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>where</mtext> </mrow> <mtext> </mtext> <mi>tan</mi> <mo>⁡</mo> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sinh</mi> <mo>⁡</mo> <mi>z</mi> </mrow> <mrow> <mi>cosh</mi> <mo>⁡</mo> <mi>w</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mi>tan</mi> <mo>⁡</mo> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sinh</mi> <mo>⁡</mo> <mi>w</mi> </mrow> <mrow> <mi>cosh</mi> <mo>⁡</mo> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>where</mtext> </mrow> <mtext> </mtext> <mi>tanh</mi> <mo>⁡</mo> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>z</mi> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>w</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mi>tanh</mi> <mo>⁡</mo> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>w</mi> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {gd} (z+w)&=u+v,\quad {\text{where}}\ \tan u={\frac {\sinh z}{\cosh w}},\ \tan v={\frac {\sinh w}{\cosh z}},\\[10mu]\operatorname {gd} ^{-1}(z+w)&=u+v,\quad {\text{where}}\ \tanh u={\frac {\sin z}{\cos w}},\ \tanh v={\frac {\sin w}{\cos z}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e97dc7f1fbf80a7d2442717388919973208aa0ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:64.731ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {gd} (z+w)&=u+v,\quad {\text{where}}\ \tan u={\frac {\sinh z}{\cosh w}},\ \tan v={\frac {\sinh w}{\cosh z}},\\[10mu]\operatorname {gd} ^{-1}(z+w)&=u+v,\quad {\text{where}}\ \tanh u={\frac {\sin z}{\cos w}},\ \tanh v={\frac {\sin w}{\cos z}},\end{aligned}}}"></span></dd></dl> <p>which can be used to derive the <a class="mw-selflink-fragment" href="#Complex_values">per-component computation</a> for the complex Gudermannian and inverse Gudermannian.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>In the specific case <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="{\textstyle z=w,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>w</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle z=w,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e56d9cd9e21a79d8e67571975e3b2e6b135ff63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.498ex; height:2.009ex;" alt="{\textstyle z=w,}"></span> double-argument identities are </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 {\begin{aligned}\operatorname {gd} (2z)&=2\arctan(\sin(\operatorname {gd} z)),\\[5mu]\operatorname {gd} ^{-1}(2z)&=2\operatorname {artanh} (\sinh(\operatorname {gd} ^{-1}z)).\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="0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>gd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>artanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {gd} (2z)&=2\arctan(\sin(\operatorname {gd} z)),\\[5mu]\operatorname {gd} ^{-1}(2z)&=2\operatorname {artanh} (\sinh(\operatorname {gd} ^{-1}z)).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a7f2451e5d3d5827bc599e1c3d711795d73c24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:35.65ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {gd} (2z)&=2\arctan(\sin(\operatorname {gd} z)),\\[5mu]\operatorname {gd} ^{-1}(2z)&=2\operatorname {artanh} (\sinh(\operatorname {gd} ^{-1}z)).\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Taylor_series">Taylor series</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gudermannian_function&action=edit&section=7" title="Edit section: Taylor series"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> near zero, valid for complex values <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="{\textstyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a33e37010e3acdeeb80fdb95df9bfe411fd79e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\textstyle z}"></span> 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="{\textstyle |z|<{\tfrac {1}{2}}\pi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle |z|<{\tfrac {1}{2}}\pi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/766bcb8ac76422ca5a8466525dd459757489ef8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.117ex; height:3.509ex;" alt="{\textstyle |z|<{\tfrac {1}{2}}\pi ,}"></span> are<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </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 {\begin{aligned}\operatorname {gd} z&=\sum _{k=0}^{\infty }{\frac {E_{k}}{(k+1)!}}z^{k+1}=z-{\frac {1}{6}}z^{3}+{\frac {1}{24}}z^{5}-{\frac {61}{5040}}z^{7}+{\frac {277}{72576}}z^{9}-\dots ,\\[10mu]\operatorname {gd} ^{-1}z&=\sum _{k=0}^{\infty }{\frac {|E_{k}|}{(k+1)!}}z^{k+1}=z+{\frac {1}{6}}z^{3}+{\frac {1}{24}}z^{5}+{\frac {61}{5040}}z^{7}+{\frac {277}{72576}}z^{9}+\dots ,\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="0.856em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>gd</mi> <mo>⁡</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>61</mn> <mn>5040</mn> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>277</mn> <mn>72576</mn> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> <mo>−</mo> <mo>…</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>61</mn> <mn>5040</mn> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>277</mn> <mn>72576</mn> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> <mo>+</mo> <mo>…</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {gd} z&=\sum _{k=0}^{\infty }{\frac {E_{k}}{(k+1)!}}z^{k+1}=z-{\frac {1}{6}}z^{3}+{\frac {1}{24}}z^{5}-{\frac {61}{5040}}z^{7}+{\frac {277}{72576}}z^{9}-\dots ,\\[10mu]\operatorname {gd} ^{-1}z&=\sum _{k=0}^{\infty }{\frac {|E_{k}|}{(k+1)!}}z^{k+1}=z+{\frac {1}{6}}z^{3}+{\frac {1}{24}}z^{5}+{\frac {61}{5040}}z^{7}+{\frac {277}{72576}}z^{9}+\dots ,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c867f2c3e83d3142bc5de97d045c1311b0d30203" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:74.659ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {gd} z&=\sum _{k=0}^{\infty }{\frac {E_{k}}{(k+1)!}}z^{k+1}=z-{\frac {1}{6}}z^{3}+{\frac {1}{24}}z^{5}-{\frac {61}{5040}}z^{7}+{\frac {277}{72576}}z^{9}-\dots ,\\[10mu]\operatorname {gd} ^{-1}z&=\sum _{k=0}^{\infty }{\frac {|E_{k}|}{(k+1)!}}z^{k+1}=z+{\frac {1}{6}}z^{3}+{\frac {1}{24}}z^{5}+{\frac {61}{5040}}z^{7}+{\frac {277}{72576}}z^{9}+\dots ,\end{aligned}}}"></span></dd></dl> <p>where the numbers <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="{\textstyle E_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle E_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62abd82e78906093d1d81a06bc46e926c35ab0fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.804ex; height:2.509ex;" alt="{\textstyle E_{k}}"></span> are the <a href="/wiki/Euler_numbers" title="Euler numbers">Euler secant numbers</a>, 1, 0, -1, 0, 5, 0, -61, 0, 1385 ... (sequences <a href="//oeis.org/A122045" class="extiw" title="oeis:A122045">A122045</a>, <a href="//oeis.org/A000364" class="extiw" title="oeis:A000364">A000364</a>, and <a href="//oeis.org/A028296" class="extiw" title="oeis:A028296">A028296</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). These series were first computed by <a href="/wiki/James_Gregory_(mathematician)" title="James Gregory (mathematician)">James Gregory</a> in 1671.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>Because the Gudermannian and inverse Gudermannian functions are the integrals of the hyperbolic secant and secant functions, the numerators <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="{\textstyle E_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle E_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62abd82e78906093d1d81a06bc46e926c35ab0fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.804ex; height:2.509ex;" alt="{\textstyle E_{k}}"></span> 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="{\textstyle |E_{k}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle |E_{k}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/313e5e861810b966807e07241b67a9416e0c1345" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.098ex; height:2.843ex;" alt="{\textstyle |E_{k}|}"></span> are same as the numerators of the <a href="/wiki/Hyperbolic_functions#Taylor_series_expressions" title="Hyperbolic functions">Taylor series for <span class="texhtml">sech</span></a> and <a href="/wiki/Trigonometric_functions#Power_series_expansion" title="Trigonometric functions"><span class="texhtml">sec</span></a>, respectively, but shifted by one place. </p><p>The reduced unsigned numerators are 1, 1, 1, 61, 277, ... and the reduced denominators are 1, 6, 24, 5040, 72576, ... (sequences <a href="//oeis.org/A091912" class="extiw" title="oeis:A091912">A091912</a> and <a href="//oeis.org/A136606" class="extiw" title="oeis:A136606">A136606</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gudermannian_function&action=edit&section=8" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">For broader coverage of this topic, see <a href="/wiki/Mercator_projection#History" title="Mercator projection">Mercator projection § History</a>, and <a href="/wiki/Integral_of_the_secant_function" title="Integral of the secant function">Integral of the secant function</a>.</div> <p>The function and its inverse are related to the <a href="/wiki/Mercator_projection" title="Mercator projection">Mercator projection</a>. The vertical coordinate in the Mercator projection is called <a href="/wiki/Latitude#Isometric_latitude" title="Latitude">isometric latitude</a>, and is often denoted <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="{\textstyle \psi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f441ae5145c06b5b873eef121d91d3efdd75f35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.16ex; height:2.509ex;" alt="{\textstyle \psi .}"></span> In terms of <a href="/wiki/Latitude" title="Latitude">latitude</a> <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="{\textstyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b36dc4887c86d6f01183292a5d2f984e23318c10" 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="{\textstyle \phi }"></span> on the sphere (expressed in <a href="/wiki/Radian" title="Radian">radians</a>) the isometric latitude can be written </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 \psi =\operatorname {gd} ^{-1}\phi =\int _{0}^{\phi }\sec t\,\mathrm {d} t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msubsup> <mi>sec</mi> <mo>⁡</mo> <mi>t</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =\operatorname {gd} ^{-1}\phi =\int _{0}^{\phi }\sec t\,\mathrm {d} t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a920098bfd9d3754f6573874769b15d15710c40e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.707ex; height:6.343ex;" alt="{\displaystyle \psi =\operatorname {gd} ^{-1}\phi =\int _{0}^{\phi }\sec t\,\mathrm {d} t.}"></span></dd></dl> <p>The inverse from the isometric latitude to spherical latitude is <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="{\textstyle \phi =\operatorname {gd} \psi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mi>gd</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi =\operatorname {gd} \psi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4094fcc850b702061e791c0637d9d53f0a7e5608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.486ex; height:2.509ex;" alt="{\textstyle \phi =\operatorname {gd} \psi .}"></span> (Note: on an <a href="/wiki/Ellipsoid_of_revolution" class="mw-redirect" title="Ellipsoid of revolution">ellipsoid of revolution</a>, the relation between geodetic latitude and isometric latitude is slightly more complicated.) </p><p><a href="/wiki/Gerardus_Mercator" title="Gerardus Mercator">Gerardus Mercator</a> plotted his celebrated map in 1569, but the precise method of construction was not revealed. In 1599, <a href="/wiki/Edward_Wright_(mathematician)" title="Edward Wright (mathematician)">Edward Wright</a> described a method for constructing a Mercator projection numerically from trigonometric tables, but did not produce a closed formula. The closed formula was published in 1668 by <a href="/wiki/James_Gregory_(mathematician)" title="James Gregory (mathematician)">James Gregory</a>. </p><p>The Gudermannian function per se was introduced by <a href="/wiki/Johann_Heinrich_Lambert" title="Johann Heinrich Lambert">Johann Heinrich Lambert</a> in the 1760s at the same time as the <a href="/wiki/Hyperbolic_functions" title="Hyperbolic functions">hyperbolic functions</a>. He called it the "transcendent angle", and it went by various names until 1862 when <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a> suggested it be given its current name as a tribute to <a href="/wiki/Christoph_Gudermann" title="Christoph Gudermann">Christoph Gudermann</a>'s work in the 1830s on the theory of special functions.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Gudermann had published articles in <i><a href="/wiki/Crelle%27s_Journal" title="Crelle's Journal">Crelle's Journal</a></i> that were later collected in a book<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> which expounded <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="{\textstyle \sinh }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>sinh</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sinh }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f54dc39a77cadfbe93dff7a9ab383194091c2513" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.148ex; height:2.176ex;" alt="{\textstyle \sinh }"></span> 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="{\textstyle \cosh }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>cosh</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \cosh }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3419abb2bed522294ca025fb5ffd9fff9d7d0bdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.404ex; height:2.176ex;" alt="{\textstyle \cosh }"></span> to a wide audience (although represented by the symbols <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="{\textstyle {\mathfrak {Sin}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">S</mi> <mi mathvariant="fraktur">i</mi> <mi mathvariant="fraktur">n</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathfrak {Sin}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecefb7a444fdbe6d637755859e58195c9995e293" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.801ex; height:2.176ex;" alt="{\textstyle {\mathfrak {Sin}}}"></span> 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="{\textstyle {\mathfrak {Cos}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">C</mi> <mi mathvariant="fraktur">o</mi> <mi mathvariant="fraktur">s</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathfrak {Cos}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c823e02b6f2d73871c88e697d261f802867cc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.592ex; height:2.176ex;" alt="{\textstyle {\mathfrak {Cos}}}"></span>). </p><p>The notation <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="{\textstyle \operatorname {gd} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>gd</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {gd} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e8496c6ca1328821c3f42dd1173a63d6dacbd05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\textstyle \operatorname {gd} }"></span> was introduced by Cayley who starts by calling <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="{\textstyle \phi =\operatorname {gd} u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mi>gd</mi> <mo>⁡</mo> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi =\operatorname {gd} u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed00c253b5012aa6edefa172488beb2c8ed663c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.656ex; height:2.509ex;" alt="{\textstyle \phi =\operatorname {gd} u}"></span> the <a href="/wiki/Jacobi_elliptic_functions#am" title="Jacobi elliptic functions">Jacobi elliptic amplitude</a> <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="{\textstyle \operatorname {am} u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>am</mi> <mo>⁡</mo> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {am} u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c8109fe0f7e018c9f4c8b61cd0ee683a4a1efd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.815ex; height:1.676ex;" alt="{\textstyle \operatorname {am} u}"></span> in the degenerate case where the elliptic modulus is <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="{\textstyle m=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle m=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90c13cd3deaf3bd232b483e7a04df890d0cdd84c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.948ex; height:2.509ex;" alt="{\textstyle m=1,}"></span> so 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="{\textstyle {\sqrt {1+m\sin \!^{2}\,\phi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mi>m</mi> <mi>sin</mi> <msup> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>ϕ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {1+m\sin \!^{2}\,\phi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d2aa9a307e0472990f477bff23377933b3babb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.436ex; height:3.509ex;" alt="{\textstyle {\sqrt {1+m\sin \!^{2}\,\phi }}}"></span> reduces to <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="{\textstyle \cos \phi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \cos \phi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe802eb64df25a701e15ba2b699ef62981470b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.53ex; height:2.509ex;" alt="{\textstyle \cos \phi .}"></span><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> This is the inverse of the <a href="/wiki/Integral_of_the_secant_function" title="Integral of the secant function">integral of the secant function</a>. Using Cayley's notation, </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 u=\int _{0}{\frac {d\phi }{\cos \phi }}={\log \,\tan }{\bigl (}{\tfrac {1}{4}}\pi +{\tfrac {1}{2}}\phi {\bigr )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ϕ</mi> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>log</mi> <mspace width="thinmathspace" /> <mi>tan</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=\int _{0}{\frac {d\phi }{\cos \phi }}={\log \,\tan }{\bigl (}{\tfrac {1}{4}}\pi +{\tfrac {1}{2}}\phi {\bigr )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8fc69dc7a8fcb5c4f28ead6392400cabe2041c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.737ex; height:5.843ex;" alt="{\displaystyle u=\int _{0}{\frac {d\phi }{\cos \phi }}={\log \,\tan }{\bigl (}{\tfrac {1}{4}}\pi +{\tfrac {1}{2}}\phi {\bigr )}.}"></span></dd></dl> <p>He then derives "the definition of the transcendent", </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 \operatorname {gd} u={{\frac {1}{i}}\log \,\tan }{\bigl (}{\tfrac {1}{4}}\pi +{\tfrac {1}{2}}ui{\bigr )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>gd</mi> <mo>⁡</mo> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>i</mi> </mfrac> </mrow> <mi>log</mi> <mspace width="thinmathspace" /> <mi>tan</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>u</mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gd} u={{\frac {1}{i}}\log \,\tan }{\bigl (}{\tfrac {1}{4}}\pi +{\tfrac {1}{2}}ui{\bigr )},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e97e04d4db5470015af2754956b88596bc806cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.159ex; height:5.176ex;" alt="{\displaystyle \operatorname {gd} u={{\frac {1}{i}}\log \,\tan }{\bigl (}{\tfrac {1}{4}}\pi +{\tfrac {1}{2}}ui{\bigr )},}"></span></dd></dl> <p>observing that "although exhibited in an imaginary form, [it] is a real function of <span class="nowrap"><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="{\textstyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24e12e26e505ed5b02c7648a89bbc6737038b2de" 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="{\textstyle u}"></span>".</span> </p><p>The Gudermannian and its inverse were used to make <a href="/wiki/Trigonometric_tables" title="Trigonometric tables">trigonometric tables</a> of circular functions also function as tables of hyperbolic functions. Given a hyperbolic angle <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="{\textstyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d738e3571903ec4e786923ddbd817cd147cb5a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\textstyle \psi }"></span>, hyperbolic functions could be found by first looking up <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="{\textstyle \phi =\operatorname {gd} \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mi>gd</mi> <mo>⁡</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi =\operatorname {gd} \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a90392c8dc8bdc35d51f9b99dd6d134d4300a4b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.839ex; height:2.509ex;" alt="{\textstyle \phi =\operatorname {gd} \psi }"></span> in a Gudermannian table and then looking up the appropriate circular function 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="{\textstyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b36dc4887c86d6f01183292a5d2f984e23318c10" 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="{\textstyle \phi }"></span>, or by directly locating <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="{\textstyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d738e3571903ec4e786923ddbd817cd147cb5a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\textstyle \psi }"></span> in an auxiliary <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 \operatorname {gd} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gd} ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a79e1a6ee5398ea160393c746ff96902f6857fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.788ex; height:3.009ex;" alt="{\displaystyle \operatorname {gd} ^{-1}}"></span> column of the trigonometric table.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Generalization">Generalization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gudermannian_function&action=edit&section=9" title="Edit section: Generalization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Gudermannian function can be thought of mapping points on one branch of a hyperbola to points on a semicircle. Points on one sheet of an <i>n</i>-dimensional <a href="/wiki/Hyperboloid" title="Hyperboloid">hyperboloid of two sheets</a> can be likewise mapped onto a <i>n</i>-dimensional hemisphere via stereographic projection. The <a href="/wiki/Hyperbolic_geometry#The_hemisphere_model" title="Hyperbolic geometry">hemisphere model of hyperbolic space</a> uses such a map to represent hyperbolic space. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gudermannian_function&action=edit&section=10" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Distance_in_the_half-plane_model_3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Distance_in_the_half-plane_model_3.png/330px-Distance_in_the_half-plane_model_3.png" decoding="async" width="330" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Distance_in_the_half-plane_model_3.png/495px-Distance_in_the_half-plane_model_3.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Distance_in_the_half-plane_model_3.png/660px-Distance_in_the_half-plane_model_3.png 2x" data-file-width="1654" data-file-height="828" /></a><figcaption>Distance in the <a href="/wiki/Poincar%C3%A9_half-plane_model" title="Poincaré half-plane model">Poincaré half-plane model</a> of the <a href="/wiki/Hyperbolic_plane" class="mw-redirect" title="Hyperbolic plane">hyperbolic plane</a> from the apex of a semicircle to another point on it is the inverse Gudermannian function of the central angle.</figcaption></figure> <ul><li>The <a href="/wiki/Angle_of_parallelism" title="Angle of parallelism">angle of parallelism</a> function in <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a> is the <a href="/wiki/Angle#Combining_angle_pairs" title="Angle">complement</a> of the gudermannian, <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 {\mathit {\Pi }}(\psi )={\tfrac {1}{2}}\pi -\operatorname {gd} \psi .}"> <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-mathit" mathvariant="italic">Π</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <mo>−</mo> <mi>gd</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {\Pi }}(\psi )={\tfrac {1}{2}}\pi -\operatorname {gd} \psi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d69561be2de197f573d58ce3bc54e48999d5f625" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.98ex; height:3.509ex;" alt="{\displaystyle {\mathit {\Pi }}(\psi )={\tfrac {1}{2}}\pi -\operatorname {gd} \psi .}"></span></li> <li>On a <a href="/wiki/Mercator_projection" title="Mercator projection">Mercator projection</a> a line of constant latitude is parallel to the equator (on the projection) at a distance proportional to the anti-gudermannian of the latitude.</li> <li>The Gudermannian function (with a complex argument) may be used to define the <a href="/wiki/Transverse_Mercator_projection" title="Transverse Mercator projection">transverse Mercator projection</a>.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li>The Gudermannian function appears in a non-periodic solution of the <a href="/wiki/Inverted_pendulum" title="Inverted pendulum">inverted pendulum</a>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></li> <li>The Gudermannian function appears in a moving mirror solution of the dynamical <a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a>.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup></li> <li>If an infinite number of infinitely long, equidistant, parallel, coplanar, straight wires are kept at equal <a href="/wiki/Electric_potential" title="Electric potential">potentials</a> with alternating signs, the potential-flux distribution in a cross-sectional plane perpendicular to the wires is the complex Gudermannian function.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></li> <li>The Gudermannian function is a <a href="/wiki/Sigmoid_function" title="Sigmoid function">sigmoid function</a>, and as such is sometimes used as an <a href="/wiki/Activation_function" title="Activation function">activation function</a> in machine learning.</li> <li>The (scaled and shifted) Gudermannian function is the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of the <a href="/wiki/Hyperbolic_secant_distribution" title="Hyperbolic secant distribution">hyperbolic secant distribution</a>.</li> <li>A function based on the Gudermannian provides a good model for the shape of <a href="/wiki/Spiral_galaxy" title="Spiral galaxy">spiral galaxy</a> arms.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></li></ul> <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=Gudermannian_function&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Tractrix" title="Tractrix">Tractrix</a></li> <li><a href="/wiki/Catenary#Catenary_of_equal_strength" title="Catenary">Catenary § Catenary of equal strength</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gudermannian_function&action=edit&section=12" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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: 25em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">The symbols <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="{\textstyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d738e3571903ec4e786923ddbd817cd147cb5a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\textstyle \psi }"></span> 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="{\textstyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b36dc4887c86d6f01183292a5d2f984e23318c10" 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="{\textstyle \phi }"></span> were chosen for this article because they are commonly used in <a href="/wiki/Geodesy" title="Geodesy">geodesy</a> for the <a href="/wiki/Latitude#Isometric_latitude" title="Latitude">isometric latitude</a> (vertical coordinate of the <a href="/wiki/Mercator_projection" title="Mercator projection">Mercator projection</a>) and <a href="/wiki/Geodetic_coordinates" title="Geodetic coordinates">geodetic latitude</a>, respectively, and geodesy/cartography was the original context for the study of the Gudermannian and inverse Gudermannian functions.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Gudermann published several papers about the trigonometric and hyperbolic functions in <a href="/wiki/Crelle%27s_Journal" title="Crelle's Journal">Crelle's Journal</a> in 1830–1831. These were collected in a book, <a href="#CITEREFGudermann1833">Gudermann (1833)</a>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFRoyOlver2010">Roy & Olver (2010)</a> <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/4.23#viii">§4.23(viii) "Gudermannian Function"</a>; <a href="#CITEREFBeyer1987">Beyer (1987)</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFKennelly1929">Kennelly (1929)</a>; <a href="#CITEREFLee1976">Lee (1976)</a></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFMasson2021">Masson (2021)</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFGottschalk2003">Gottschalk (2003)</a> pp. 23–27</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFMasson2021">Masson (2021)</a> draws complex-valued plots of several of these, demonstrating that naïve implementations that choose the principal branch of inverse trigonometric functions yield incorrect results.</span> </li> <li id="cite_note-weinstein-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-weinstein_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-weinstein_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Gudermannian"><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="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/Gudermannian.html">"Gudermannian"</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=Gudermannian&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FGudermannian.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGudermannian+function" class="Z3988"></span></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="#CITEREFKennelly1929">Kennelly (1929)</a></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFKennelly1929">Kennelly (1929)</a> <a rel="nofollow" class="external text" href="https://archive.org/details/dli.ministry.19102/page/181">p. 181</a>; <a href="#CITEREFBeyer1987">Beyer (1987)</a> <a rel="nofollow" class="external text" href="https://archive.org/details/crchandbookofmat00beye/page/269/mode/1up">p. 269</a></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="#CITEREFBeyer1987">Beyer (1987)</a> <a rel="nofollow" class="external text" href="https://archive.org/details/crchandbookofmat00beye/page/269/mode/1up">p. 269</a> – note the typo.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFLegendre1817">Legendre (1817)</a> <a rel="nofollow" class="external text" href="https://archive.org/details/exercicescalculi02legerich/page/n165/">§4.2.8(163) pp. 144–145</a></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#CITEREFKennelly1929">Kennelly (1929)</a> <a rel="nofollow" class="external text" href="https://archive.org/details/dli.ministry.19102/page/182">p. 182</a></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="#CITEREFKahligReich2013">Kahlig & Reich (2013)</a></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a href="#CITEREFCayley1862">Cayley (1862)</a> <a rel="nofollow" class="external text" href="https://archive.org/details/londonedinburg4241862lond/page/21">p. 21</a></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a href="#CITEREFKennelly1929">Kennelly (1929)</a> <a rel="nofollow" class="external text" href="https://archive.org/details/dli.ministry.19102/page/180">pp. 180–183</a></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><a href="#CITEREFLegendre1817">Legendre (1817)</a> <a rel="nofollow" class="external text" href="https://archive.org/details/exercicescalculi02legerich/page/n165/">§4.2.7(162) pp. 143–144</a></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTurnbull1939" class="citation book cs1">Turnbull, Herbert Westren, ed. (1939). <i>James Gregory; Tercentenary Memorial Volume</i>. G. Bell & Sons. p. 170.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=James+Gregory%3B+Tercentenary+Memorial+Volume&rft.pages=170&rft.pub=G.+Bell+%26+Sons&rft.date=1939&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGudermannian+function" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><a href="#CITEREFBeckerVan_Orstrand1909">Becker & Van Orstrand (1909)</a></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a href="#CITEREFGudermann1833">Gudermann (1833)</a></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><a href="#CITEREFCayley1862">Cayley (1862)</a></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">For example Hoüel labels the hyperbolic functions across the top in Table XIV of: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHoüel1885" class="citation book cs1">Hoüel, Guillaume Jules (1885). <a rel="nofollow" class="external text" href="https://archive.org/details/recueildeformul00hogoog/page/n115/"><i>Recueil de formules et de tables numériques</i></a>. Gauthier-Villars. p. 36.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Recueil+de+formules+et+de+tables+num%C3%A9riques&rft.pages=36&rft.pub=Gauthier-Villars&rft.date=1885&rft.aulast=Ho%C3%BCel&rft.aufirst=Guillaume+Jules&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Frecueildeformul00hogoog%2Fpage%2Fn115%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGudermannian+function" class="Z3988"></span> </span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><a href="#CITEREFOsborne2013">Osborne (2013)</a> p. 74</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><a href="#CITEREFRobertson1997">Robertson (1997)</a></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><a href="#CITEREFGoodAndersonEvans2013">Good, Anderson & Evans (2013)</a></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><a href="#CITEREFKennelly1928">Kennelly (1928)</a></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><a href="#CITEREFRingermacherMead2009">Ringermacher & Mead (2009)</a></span> </li> </ol></div> <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=Gudermannian_function&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarnett2004" class="citation journal cs1"><a href="/wiki/Janet_Barnett" title="Janet Barnett">Barnett, Janet Heine</a> (2004). <a rel="nofollow" class="external text" href="https://www.maa.org/sites/default/files/321922717729.pdf.bannered.pdf">"Enter, Stage Center: The Early Drama of the Hyperbolic Functions"</a> <span class="cs1-format">(PDF)</span>. <i>Mathematics Magazine</i>. <b>77</b> (1): <span class="nowrap">15–</span>30. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F0025570X.2004.11953223">10.1080/0025570X.2004.11953223</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Enter%2C+Stage+Center%3A+The+Early+Drama+of+the+Hyperbolic+Functions&rft.volume=77&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E15-%3C%2Fspan%3E30&rft.date=2004&rft_id=info%3Adoi%2F10.1080%2F0025570X.2004.11953223&rft.aulast=Barnett&rft.aufirst=Janet+Heine&rft_id=https%3A%2F%2Fwww.maa.org%2Fsites%2Fdefault%2Ffiles%2F321922717729.pdf.bannered.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGudermannian+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeckerVan_Orstrand1909" class="citation book cs1"><a href="/wiki/George_Ferdinand_Becker" title="George Ferdinand Becker">Becker, George Ferdinand</a>; Van Orstrand, Charles Edwin (1909). <a rel="nofollow" class="external text" href="https://archive.org/details/smithsonianmathe00smituoft"><i>Hyperbolic Functions</i></a>. Smithsonian Mathematical Tables. Smithsonian Institution.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hyperbolic+Functions&rft.series=Smithsonian+Mathematical+Tables&rft.pub=Smithsonian+Institution&rft.date=1909&rft.aulast=Becker&rft.aufirst=George+Ferdinand&rft.au=Van+Orstrand%2C+Charles+Edwin&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsmithsonianmathe00smituoft&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGudermannian+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBecker1912" class="citation journal cs1">Becker, George Ferdinand (1912). <a rel="nofollow" class="external text" href="https://ia600708.us.archive.org/view_archive.php?archive=/28/items/crossref-pre-1923-scholarly-works/10.1080%252F14786440908635958.zip&file=10.1080%252F14786441008637363.pdf">"The gudermannian complement and imaginary geometry"</a> <span class="cs1-format">(PDF)</span>. <i>The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science</i>. <b>24</b> (142): <span class="nowrap">600–</span>608. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786441008637363">10.1080/14786441008637363</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+London%2C+Edinburgh%2C+and+Dublin+Philosophical+Magazine+and+Journal+of+Science&rft.atitle=The+gudermannian+complement+and+imaginary+geometry&rft.volume=24&rft.issue=142&rft.pages=%3Cspan+class%3D%22nowrap%22%3E600-%3C%2Fspan%3E608&rft.date=1912&rft_id=info%3Adoi%2F10.1080%2F14786441008637363&rft.aulast=Becker&rft.aufirst=George+Ferdinand&rft_id=https%3A%2F%2Fia600708.us.archive.org%2Fview_archive.php%3Farchive%3D%2F28%2Fitems%2Fcrossref-pre-1923-scholarly-works%2F10.1080%25252F14786440908635958.zip%26file%3D10.1080%25252F14786441008637363.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGudermannian+function" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeyer1987" class="citation book cs1">Beyer, William H., ed. (1987). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/crchandbookofmat00beye/"><i>CRC Handbook of Mathematical Sciences</i></a></span> (6th ed.). 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Gudermannian function - Wikipedia