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Gudermannian function

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In mathematics, the Gudermannian function relates a <a href="Hyperbolic_angle" title="Hyperbolic angle">hyperbolic angle</a> measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \psi }">

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<annotation encoding="application/x-tex">{\textstyle \psi }</annotation>

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</math><img src="./9d738e3571903ec4e786923ddbd817cd147cb5a2.svg" 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 }" loading="lazy"> to a <a href="Angle" title="Angle">circular angle</a> measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi }">

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<annotation encoding="application/x-tex">{\textstyle \phi }</annotation>

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</math><img src="./b36dc4887c86d6f01183292a5d2f984e23318c10.svg" 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 }" loading="lazy"> called the gudermannian of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \psi }">

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<annotation encoding="application/x-tex">{\textstyle \psi }</annotation>

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</math><img src="./9d738e3571903ec4e786923ddbd817cd147cb5a2.svg" 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 }" loading="lazy"> and denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {gd} \psi }">

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<annotation encoding="application/x-tex">{\textstyle \operatorname {gd} \psi }</annotation>

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</math><img src="./d7833c345c8798d7f6a5b4401d71a3f1a6f8d7b7.svg" 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 }" loading="lazy">.<a href="#cite_note-1">[1]</a> The Gudermannian function reveals a close relationship between the <a href="Circular_function" class="mw-redirect" title="Circular function">circular functions</a> and <a href="Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">hyperbolic functions</a>. It was introduced in the 1760s by <a href="Johann_Heinrich_Lambert" title="Johann Heinrich Lambert">Johann Heinrich Lambert</a>, and later named for <a href="Christoph_Gudermann" title="Christoph Gudermann">Christoph Gudermann</a> who also described the relationship between circular and hyperbolic functions in 1830.<a href="#cite_note-2">[2]</a> The gudermannian is sometimes called the hyperbolic amplitude as a <a href="Limiting_case_(mathematics)" title="Limiting case (mathematics)">limiting case</a> of the <a href="Jacobi_elliptic_functions#am" title="Jacobi elliptic functions">Jacobi elliptic amplitude</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {am} (\psi ,m)}">

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<annotation encoding="application/x-tex">{\textstyle \operatorname {am} (\psi ,m)}</annotation>

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</math><img src="./bb75dec9c8aa0e577f2f558db0f83dae9f2ef77e.svg" 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)}" loading="lazy"> when parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle m=1.}">

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<annotation encoding="application/x-tex">{\textstyle m=1.}</annotation>

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</math><img src="./25d81740fd298793563fa19a8e0ed08fdd89738e.svg" 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.}" loading="lazy">

The <a href="Real_number" title="Real number">real</a> Gudermannian function is typically defined for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle -\infty <\psi <\infty }">

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<annotation encoding="application/x-tex">{\textstyle -\infty <\psi <\infty }</annotation>

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</math><img src="./e75be30118b3e1539d58c04f169ac31b90ed3a29.svg" 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 }" loading="lazy"> to be the integral of the hyperbolic secant<a href="#cite_note-3">[3]</a>

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<mi>arctan</mi>

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<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>

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</math><img src="./bda4b2ebdd63e0a668c77c86cf2075b2effa2832.svg" 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 ).}" loading="lazy"></dd></dl>

The real inverse Gudermannian function can be defined for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle -{\tfrac {1}{2}}\pi <\phi <{\tfrac {1}{2}}\pi }">

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<annotation encoding="application/x-tex">{\textstyle -{\tfrac {1}{2}}\pi <\phi <{\tfrac {1}{2}}\pi }</annotation>

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</math><img src="./6a3ae0f4df0531eaba3c0a793974483285508471.svg" 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 }" loading="lazy"> as the <a href="Integral_of_the_secant_function" title="Integral of the secant function">integral of the (circular) secant</a>

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<mi>arsinh</mi>

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<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>

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</math><img src="./76c0302e0820820cf3804ab0b839f12d330716d2.svg" 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 ).}" loading="lazy"></dd></dl>

The hyperbolic angle measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =\operatorname {gd} ^{-1}\phi }">

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<annotation encoding="application/x-tex">{\displaystyle \psi =\operatorname {gd} ^{-1}\phi }</annotation>

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</math><img src="./82961cdf95e4e7a4f3b2077f565baf68b60742dd.svg" 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 }" loading="lazy"> is called the anti-gudermannian of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

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</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" 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 }" loading="lazy"> or sometimes the lambertian of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

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</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" 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 }" loading="lazy">, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =\operatorname {lam} \phi .}">

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<annotation encoding="application/x-tex">{\displaystyle \psi =\operatorname {lam} \phi .}</annotation>

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</math><img src="./430b2e5e131383a2f2c52c06f67f82588bb4a61d.svg" 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 .}" loading="lazy"><a href="#cite_note-4">[4]</a> In the context of <a href="Geodesy" title="Geodesy">geodesy</a> and <a href="Navigation" title="Navigation">navigation</a> for latitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi }">

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<annotation encoding="application/x-tex">{\textstyle \phi }</annotation>

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</math><img src="./b36dc4887c86d6f01183292a5d2f984e23318c10.svg" 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 }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\operatorname {gd} ^{-1}\phi }">

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<annotation encoding="application/x-tex">{\displaystyle k\operatorname {gd} ^{-1}\phi }</annotation>

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</math><img src="./b9a654a3f871e362330c6c00df4dcf088bdb4143.svg" 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 }" loading="lazy"> (scaled by arbitrary constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k}">

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<annotation encoding="application/x-tex">{\textstyle k}</annotation>

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</math><img src="./0d5595fc0c47452f8fc2aa6e29c3611f047714b0.svg" 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}" loading="lazy">) was historically called the meridional part of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

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</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" 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 }" loading="lazy"> (<a href="French_(language)" class="mw-redirect" title="French (language)">French</a>: latitude croissante). It is the vertical coordinate of the <a href="Mercator_projection" title="Mercator projection">Mercator projection</a>.

The two angle measures <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi }">

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<annotation encoding="application/x-tex">{\textstyle \phi }</annotation>

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</math><img src="./b36dc4887c86d6f01183292a5d2f984e23318c10.svg" 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 }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \psi }">

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<annotation encoding="application/x-tex">{\textstyle \psi }</annotation>

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</math><img src="./9d738e3571903ec4e786923ddbd817cd147cb5a2.svg" 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 }" loading="lazy"> are related by a common <a href="Stereographic_projection" title="Stereographic projection">stereographic projection</a>

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<annotation encoding="application/x-tex">{\displaystyle s=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi ,}</annotation>

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</math><img src="./2dea5fd46ee49cac81ff9bd5960d1e46ecddaad1.svg" 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 ,}" loading="lazy"></dd></dl>

and this identity can serve as an alternative definition for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {gd} }">

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<annotation encoding="application/x-tex">{\textstyle \operatorname {gd} }</annotation>

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</math><img src="./9e8496c6ca1328821c3f42dd1173a63d6dacbd05.svg" 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} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {gd} ^{-1}}">

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<annotation encoding="application/x-tex">{\textstyle \operatorname {gd} ^{-1}}</annotation>

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</math><img src="./c67c70418c886a7bc07a9078fecbef157b393a99.svg" 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}}" loading="lazy"> valid throughout the <a href="Complex_plane" title="Complex plane">complex plane</a>:

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<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>

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</math><img src="./6dd27bd5d8e880b8a5a6bdeff8d4a27df40264c0.svg" 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}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Circular–hyperbolic_identities">Circular–hyperbolic identities</h2></div>

We can evaluate the integral of the hyperbolic secant using the stereographic projection (<a href="Tangent_half-angle_substitution#Hyperbolic_functions" title="Tangent half-angle substitution">hyperbolic half-tangent</a>) as a <a href="Integration_by_substitution" title="Integration by substitution">change of variables</a>:<a href="#cite_note-5">[5]</a>

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<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>

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</math><img src="./5945db7bc55d0b033a77986e9cc3ea1c2c529eb9.svg" 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}}}" loading="lazy"></dd></dl>

Letting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi =\operatorname {gd} \psi }">

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<annotation encoding="application/x-tex">{\textstyle \phi =\operatorname {gd} \psi }</annotation>

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</math><img src="./a90392c8dc8bdc35d51f9b99dd6d134d4300a4b0.svg" 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 }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle s=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi }">

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<annotation encoding="application/x-tex">{\textstyle s=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi }</annotation>

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</math><img src="./8df0ddd190fa55c408b835fbccf2e6c73a0e3afc.svg" 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 }" loading="lazy"> we can derive a number of identities between hyperbolic functions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \psi }">

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</math><img src="./9d738e3571903ec4e786923ddbd817cd147cb5a2.svg" 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 }" loading="lazy"> and circular functions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi .}">

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<annotation encoding="application/x-tex">{\textstyle \phi .}</annotation>

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</math><img src="./0171a72f233f82af593e5c440d7b5cd3ae388af6.svg" 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 .}" loading="lazy"><a href="#cite_note-6">[6]</a>

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</msup>

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<mrow>

<msup>

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</msup>

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</mfrac>

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<mtd>

</mtd>

<mtd>

<mfrac>

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<msup>

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</msup>

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<mrow>

<msup>

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</msup>

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</mfrac>

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<mtd>

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<mtr>

<mtd>

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<mrow>

<msup>

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</msup>

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<mtd>

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<mtd>

<mfrac>

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<msup>

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</msup>

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<mrow>

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</mfrac>

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<mtd>

</mtd>

</mtr>

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<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><img src="./3b899ada2ea0fb5c5f68d29fefb9498e1d57d2ea.svg" 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}}}" loading="lazy"></dd></dl>

These are commonly used as expressions for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gd} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {gd} }</annotation>

</semantics>

</math><img src="./6dad73bd4edce09e79f274f0bfcd574c0c32a46a.svg" 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} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gd} ^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {gd} ^{-1}}</annotation>

</semantics>

</math><img src="./1a79e1a6ee5398ea160393c746ff96902f6857fa.svg" 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}}" loading="lazy"> for real values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" 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 }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

</semantics>

</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" 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 }" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi |<{\tfrac {1}{2}}\pi .}">

</mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\phi |<{\tfrac {1}{2}}\pi .}</annotation>

</semantics>

</math><img src="./ad50d61647f0c09b281efef44887ad39c5cdab78.svg" 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 .}" loading="lazy"> For example, the numerically well-behaved formulas

<mtr>

<mtd>

</mtd>

<mtd>

<mi>arctan</mi>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<mi>arsinh</mi>

</mtd>

</mtr>

</mtable>

</mrow>

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</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><img src="./1cbb02476a2f58f8303e30dde2cd910b3357ea87.svg" 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}}}" loading="lazy"></dd></dl>

(Note, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi |>{\tfrac {1}{2}}\pi }">

</mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\phi |>{\tfrac {1}{2}}\pi }</annotation>

</semantics>

</math><img src="./8ebf58c954a4cb4f5f6588cc1ae44649ff7c455b.svg" 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 }" loading="lazy"> and for complex arguments, care must be taken choosing <a href="Branch_point" title="Branch point">branches</a> of the inverse functions.)<a href="#cite_note-7">[7]</a>

We can also express <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \psi }</annotation>

</semantics>

</math><img src="./9d738e3571903ec4e786923ddbd817cd147cb5a2.svg" 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 }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \phi }</annotation>

</semantics>

</math><img src="./b36dc4887c86d6f01183292a5d2f984e23318c10.svg" 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 }" loading="lazy"> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle s\colon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle s\colon }</annotation>

</semantics>

</math><img src="./d1f10a92e0bdd50eb65472974064e685ad708acd.svg" 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 }" loading="lazy">

<mtr>

<mtd>

<mi>arctan</mi>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<mi>artanh</mi>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</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><img src="./430ca5043a59099968092d81866f11c67f7ee130.svg" 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}}}" loading="lazy"></dd></dl>

If we expand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \tan {\tfrac {1}{2}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \tan {\tfrac {1}{2}}}</annotation>

</semantics>

</math><img src="./14c70fad71d4650185650afdb63830356769a30f.svg" 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}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \tanh {\tfrac {1}{2}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \tanh {\tfrac {1}{2}}}</annotation>

</semantics>

</math><img src="./63fa2fc97d762f87535db07dbe22273f2518f05d.svg" 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}}}" loading="lazy"> in terms of the <a href="Exponential_function#Complex_plane" title="Exponential function">exponential</a>, then we can see that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle s,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle s,}</annotation>

</semantics>

</math><img src="./d26efb726dea050d191f29941d5a1f005a3289a1.svg" 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,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp \phi i,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp \phi i,}</annotation>

</semantics>

</math><img src="./6e1a7962be704bc88bc0b81d9c22cf29f414cc91.svg" 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,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp \psi }</annotation>

</semantics>

</math><img src="./5dadd56637a0dd008b658bf676a868949f366078.svg" 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 }" loading="lazy"> are all <a href="M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a> of each-other (specifically, rotations of the <a href="Riemann_sphere" title="Riemann sphere">Riemann sphere</a>):

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<mtd>

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<mtd>

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<msup>

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</msup>

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<mfrac>

<mrow>

<msup>

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</msup>

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<mrow>

<msup>

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</msup>

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<mtr>

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<mfrac>

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<msup>

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</msup>

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<mfrac>

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</mfrac>

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<mtd>

<mfrac>

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<msup>

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</msup>

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<mrow>

<msup>

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</mfrac>

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</mtr>

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</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><img src="./1d92cae8f3cf95d4e45c8676a07e2196614fee0f.svg" 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}}}" loading="lazy"></dd></dl>

For real values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \psi }</annotation>

</semantics>

</math><img src="./9d738e3571903ec4e786923ddbd817cd147cb5a2.svg" 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 }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \phi }</annotation>

</semantics>

</math><img src="./b36dc4887c86d6f01183292a5d2f984e23318c10.svg" 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 }" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi |<{\tfrac {1}{2}}\pi }">

</mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\phi |<{\tfrac {1}{2}}\pi }</annotation>

</semantics>

</math><img src="./8f3d338d77072c0d07ad3744fe4c7cc059af8008.svg" 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 }" loading="lazy">, these Möbius transformations can be written in terms of trigonometric functions in several ways,

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<mtd>

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<mfrac>

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</mfrac>

</mrow>

<msqrt>

<mfrac>

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</mfrac>

</msqrt>

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<mtr>

<mtd>

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<mtd>

<mfrac>

</mfrac>

</mstyle>

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<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mfrac>

</mrow>

<msqrt>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

</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><img src="./f95aa3728688e9830bd9119ccae4992b11a4f703.svg" 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}}}" loading="lazy"></dd></dl>

These give further expressions for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gd} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {gd} }</annotation>

</semantics>

</math><img src="./6dad73bd4edce09e79f274f0bfcd574c0c32a46a.svg" 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} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gd} ^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {gd} ^{-1}}</annotation>

</semantics>

</math><img src="./1a79e1a6ee5398ea160393c746ff96902f6857fa.svg" 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}}" loading="lazy"> for real arguments with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi |<{\tfrac {1}{2}}\pi .}">

</mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\phi |<{\tfrac {1}{2}}\pi .}</annotation>

</semantics>

</math><img src="./ad50d61647f0c09b281efef44887ad39c5cdab78.svg" 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 .}" loading="lazy"> For example,<a href="#cite_note-weinstein-8">[8]</a>

<mtr>

<mtd>

</mtd>

<mtd>

<mi>arctan</mi>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

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</msup>

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</mtable>

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</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><img src="./e9899f14b480bfe2561cb64a139a22f14cd0c16d.svg" 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}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Complex_values">Complex values</h2></div>

As a <a href="Complex_analysis" title="Complex analysis">function of a complex variable</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle z\mapsto w=\operatorname {gd} z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle z\mapsto w=\operatorname {gd} z}</annotation>

</semantics>

</math><img src="./0bdb4144b7311fd07edc54a378e64335c2ef276e.svg" 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}" loading="lazy"> <a href="Conformal_map" title="Conformal map">conformally maps</a> the infinite strip <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left|\operatorname {Im} z\right|\leq {\tfrac {1}{2}}\pi }">

<mrow>

<mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \left|\operatorname {Im} z\right|\leq {\tfrac {1}{2}}\pi }</annotation>

</semantics>

</math><img src="./48fb80f96e3b703ba40d66546b707dee5f51bfc1.svg" 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 }" loading="lazy"> to the infinite strip <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left|\operatorname {Re} w\right|\leq {\tfrac {1}{2}}\pi ,}">

<mrow>

<mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \left|\operatorname {Re} w\right|\leq {\tfrac {1}{2}}\pi ,}</annotation>

</semantics>

</math><img src="./e8c7d6d15edeecf351d965f0adab59a04fe1c0e8.svg" 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 ,}" loading="lazy"> while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle w\mapsto z=\operatorname {gd} ^{-1}w}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle w\mapsto z=\operatorname {gd} ^{-1}w}</annotation>

</semantics>

</math><img src="./0d57e4ce6ed6e550c007a694fdfe39ec274c0f49.svg" 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}" loading="lazy"> conformally maps the infinite strip <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left|\operatorname {Re} w\right|\leq {\tfrac {1}{2}}\pi }">

<mrow>

<mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \left|\operatorname {Re} w\right|\leq {\tfrac {1}{2}}\pi }</annotation>

</semantics>

</math><img src="./552017bf9a52cb9c0ab424ce172c15a6a2b1257c.svg" 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 }" loading="lazy"> to the infinite strip <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left|\operatorname {Im} z\right|\leq {\tfrac {1}{2}}\pi .}">

<mrow>

<mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \left|\operatorname {Im} z\right|\leq {\tfrac {1}{2}}\pi .}</annotation>

</semantics>

</math><img src="./f4b53669df6286ddb122a107f08b815eed91770c.svg" 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 .}" loading="lazy">

<a href="Analytic_continuation" title="Analytic continuation">Analytically continued</a> by <a href="Schwarz_reflection_principle" title="Schwarz reflection principle">reflections</a> to the whole complex plane, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle z\mapsto w=\operatorname {gd} z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle z\mapsto w=\operatorname {gd} z}</annotation>

</semantics>

</math><img src="./0bdb4144b7311fd07edc54a378e64335c2ef276e.svg" 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}" loading="lazy"> is a periodic function of period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2\pi i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle 2\pi i}</annotation>

</semantics>

</math><img src="./92f94a26afea8d967ee55013aca00dc6c9ab3e77.svg" 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}" loading="lazy"> which sends any infinite strip of "height" <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2\pi i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle 2\pi i}</annotation>

</semantics>

</math><img src="./92f94a26afea8d967ee55013aca00dc6c9ab3e77.svg" 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}" loading="lazy"> onto the strip <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle -\pi <\operatorname {Re} w\leq \pi .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle -\pi <\operatorname {Re} w\leq \pi .}</annotation>

</semantics>

</math><img src="./7a4cd438bb7cdaf081d16d7d8b485dfe4b81cbb0.svg" 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 .}" loading="lazy"> Likewise, extended to the whole complex plane, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle w\mapsto z=\operatorname {gd} ^{-1}w}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle w\mapsto z=\operatorname {gd} ^{-1}w}</annotation>

</semantics>

</math><img src="./0d57e4ce6ed6e550c007a694fdfe39ec274c0f49.svg" 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}" loading="lazy"> is a periodic function of period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle 2\pi }</annotation>

</semantics>

</math><img src="./584b74e57567454e6a48e040ec231214be5b8c54.svg" 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 }" loading="lazy"> which sends any infinite strip of "width" <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle 2\pi }</annotation>

</semantics>

</math><img src="./584b74e57567454e6a48e040ec231214be5b8c54.svg" 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 }" loading="lazy"> onto the strip <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle -\pi <\operatorname {Im} z\leq \pi .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle -\pi <\operatorname {Im} z\leq \pi .}</annotation>

</semantics>

</math><img src="./c8436da86329e92801451943c901a52bead35eea.svg" 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 .}" loading="lazy"><a href="#cite_note-9">[9]</a> For all points in the complex plane, these functions can be correctly written as:

<mtr>

<mtd>

</mtd>

<mtd>

<mi>arctan</mi>

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</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

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</msup>

</mtd>

<mtd>

<mi>artanh</mi>

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<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

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</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><img src="./23d2333122dc1bc39541bfcaa136cacd8b8c46a0.svg" 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}}}" loading="lazy"></dd></dl>

For the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {gd} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {gd} }</annotation>

</semantics>

</math><img src="./9e8496c6ca1328821c3f42dd1173a63d6dacbd05.svg" 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} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {gd} ^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {gd} ^{-1}}</annotation>

</semantics>

</math><img src="./c67c70418c886a7bc07a9078fecbef157b393a99.svg" 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}}" loading="lazy"> functions to remain invertible with these extended domains, we might consider each to be a <a href="Multivalued_function" title="Multivalued function">multivalued function</a> (perhaps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {Gd} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {Gd} }</annotation>

</semantics>

</math><img src="./aa53c8a5b4ddec79a30cd82af850b08a2de6e190.svg" 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} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {Gd} ^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {Gd} ^{-1}}</annotation>

</semantics>

</math><img src="./2fa17b13c05f58bdf75cb68ba29430bb1bf2012e.svg" 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}}" loading="lazy">, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {gd} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {gd} }</annotation>

</semantics>

</math><img src="./9e8496c6ca1328821c3f42dd1173a63d6dacbd05.svg" 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} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {gd} ^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {gd} ^{-1}}</annotation>

</semantics>

</math><img src="./c67c70418c886a7bc07a9078fecbef157b393a99.svg" 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}}" loading="lazy"> the <a href="Principal_branch" title="Principal branch">principal branch</a>) or consider their domains and codomains as <a href="Riemann_surface" title="Riemann surface">Riemann surfaces</a>.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle u+iv=\operatorname {gd} (x+iy),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle u+iv=\operatorname {gd} (x+iy),}</annotation>

</semantics>

</math><img src="./31664a4c0b6bb4e267ab655eb14b635083c3981b.svg" 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),}" loading="lazy"> then the real and imaginary components <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle u}</annotation>

</semantics>

</math><img src="./24e12e26e505ed5b02c7648a89bbc6737038b2de.svg" 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}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle v}</annotation>

</semantics>

</math><img src="./e970b082b4c08f1583d01d713815a0900fab6a8c.svg" 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}" loading="lazy"> can be found by:<a href="#cite_note-10">[10]</a>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</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><img src="./1e4e80ae5f98b4aca39d6f4645d456efbc8b75d9.svg" 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}}.}" loading="lazy"></dd></dl>

(In practical implementation, make sure to use the <a href="Atan2" title="Atan2">2-argument arctangent</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle u=\operatorname {atan2} (\sinh x,\cos y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle u=\operatorname {atan2} (\sinh x,\cos y)}</annotation>

</semantics>

</math><img src="./e6343defd6ee31e2654a7660a846879f01f5e262.svg" 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)}" loading="lazy">.)

Likewise, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x+iy=\operatorname {gd} ^{-1}(u+iv),}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle x+iy=\operatorname {gd} ^{-1}(u+iv),}</annotation>

</semantics>

</math><img src="./f7f119c52abd01acdc70fab29445e38efaece085.svg" 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),}" loading="lazy"> then components <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle x}</annotation>

</semantics>

</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.svg" 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}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle y}</annotation>

</semantics>

</math><img src="./db9936ddb2761b76fa640fb275cb5d1fa4d6fa23.svg" 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}" loading="lazy"> can be found by:<a href="#cite_note-11">[11]</a>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</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><img src="./95bfbf84aab10fd2dbba5b857a4f916a3e2bea64.svg" 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}}.}" loading="lazy"></dd></dl>

Multiplying these together reveals the additional identity<a href="#cite_note-weinstein-8">[8]</a>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \tanh x\,\tan y=\tan u\,\tanh v.}</annotation>

</semantics>

</math><img src="./c30395d21f39b27da97ab2bfad9e860a51cb56fa.svg" 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.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Symmetries">Symmetries</h3></div>

The two functions can be thought of as rotations or reflections of each-other, with a similar relationship as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sinh iz=i\sin z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \sinh iz=i\sin z}</annotation>

</semantics>

</math><img src="./9553444ba57f192fd1fa063907a49d125ff7aec1.svg" 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}" loading="lazy"> <a href="Hyperbolic_functions#Hyperbolic_functions_for_complex_numbers" title="Hyperbolic functions">between sine and hyperbolic sine</a>:<a href="#cite_note-12">[12]</a>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

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</msup>

</mtd>

<mtd>

</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><img src="./c32f3f162617f8e2fbf080a44980b28db1dac579.svg" 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}}}" loading="lazy"></dd></dl>

The functions are both <a href="Even_and_odd_functions" title="Even and odd functions">odd</a> and they commute with <a href="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:

<mtr>

<mtd>

</mtd>

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<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>

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</math><img src="./180914d120718119ca1f5bce88c563adca935cbe.svg" 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}}}" loading="lazy"></dd></dl>

The functions are <a href="Periodic_function" title="Periodic function">periodic</a>, with periods <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2\pi i}">

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<annotation encoding="application/x-tex">{\textstyle 2\pi }</annotation>

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</math><img src="./584b74e57567454e6a48e040ec231214be5b8c54.svg" 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 }" loading="lazy">:

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<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>

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</math><img src="./1dad8e99fb169c5a17b03343210d21d41c7f19c9.svg" 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}}}" loading="lazy"></dd></dl>

A translation in the domain of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {gd} }">

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</math><img src="./9e8496c6ca1328821c3f42dd1173a63d6dacbd05.svg" 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} }" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \pm \pi i}">

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</math><img src="./6446f42e6aac3d44906105503f4b762c358665be.svg" 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}" loading="lazy"> results in a half-turn rotation and translation in the codomain by one of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \pm \pi ,}">

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</math><img src="./8dae1d4614b7b9d8367d047403284237b395d1e2.svg" 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 ,}" loading="lazy"> and vice versa for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {gd} ^{-1}\colon }">

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</math><img src="./b88a9e6ea28b014d77421776ea5148ca97eec6c8.svg" 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 }" loading="lazy"><a href="#cite_note-13">[13]</a>

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<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>

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</math><img src="./71d910f6f58789c6e15efc9e96c171b8e56ce821.svg" 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}}}" loading="lazy"></dd></dl>

A reflection in the domain of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {gd} }">

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</math><img src="./9e8496c6ca1328821c3f42dd1173a63d6dacbd05.svg" 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} }" loading="lazy"> across either of the lines <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x\pm {\tfrac {1}{2}}\pi i}">

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</math><img src="./bc59dfc13d3c5f8898b09234dce267074d4f3ab8.svg" 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}" loading="lazy"> results in a reflection in the codomain across one of the lines <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \pm {\tfrac {1}{2}}\pi +yi,}">

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</math><img src="./e92a89b566ab7cb2725ca216e17d7f3241a6153f.svg" 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,}" loading="lazy"> and vice versa for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {gd} ^{-1}\colon }">

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<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>

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</math><img src="./618b9377241278124edfde77b44a4582fdfab5d7.svg" 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}}}" loading="lazy"></dd></dl>

This is related to the identity

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<annotation encoding="application/x-tex">{\displaystyle \tanh {\tfrac {1}{2}}({\pi i}\pm z)=\tan {\tfrac {1}{2}}({\pi }\mp \operatorname {gd} z).}</annotation>

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</math><img src="./c8305fde49e12cd93b5f66d327559b45e3cff764.svg" 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).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Specific_values">Specific values</h3></div>

A few specific values (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \infty }">

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</math><img src="./7672961cb69498135a93484fe61fedd72996ad03.svg" 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 }" loading="lazy"> indicates the limit at one end of the infinite strip):<a href="#cite_note-14">[14]</a>

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<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>

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</math><img src="./62137050459c2dfe94c8ad4890d52bac8dceb818.svg" 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}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Derivatives">Derivatives</h2></div>

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:

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<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>

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</math><img src="./e5883cae1281f8cb0b8c4fb7ef64310e84ea07de.svg" 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}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Argument-addition_identities">Argument-addition identities</h2></div>

By combining <a href="Hyperbolic_functions#Sums_of_arguments" title="Hyperbolic functions">hyperbolic</a> and <a href="Trigonometric_functions#Sum_and_difference_formulas" title="Trigonometric functions">circular</a> argument-addition identities,

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<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><img src="./105177a58b358faed394dbaeb18c08c6b38f6876.svg" 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}}}" loading="lazy"></dd></dl>

with the <a href="#Circular–hyperbolic_identities">circular–hyperbolic identity</a>,

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<annotation encoding="application/x-tex">{\displaystyle \tan {\tfrac {1}{2}}(\operatorname {gd} z)=\tanh {\tfrac {1}{2}}z,}</annotation>

</semantics>

</math><img src="./5f165495d4349adf7b2cf89e469cc6669599b81a.svg" 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,}" loading="lazy"></dd></dl>

we have the Gudermannian argument-addition identities:

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</mtd>

<mtd>

<mi>arctan</mi>

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<mtd>

<mi>artanh</mi>

<mfrac>

<mrow>

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</mfrac>

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</mrow>

<msup>

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<msup>

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<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><img src="./a1c23629943ec1c0054a3d0f36066bedc0ee8cba.svg" 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}}}" loading="lazy"></dd></dl>

Further argument-addition identities can be written in terms of other circular functions,<a href="#cite_note-15">[15]</a> but they require greater care in choosing branches in inverse functions. Notably,

<mtr>

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<mtext>where</mtext>

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<mtd>

<mtext>where</mtext>

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<mfrac>

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<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><img src="./e97dc7f1fbf80a7d2442717388919973208aa0ce.svg" 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}}}" loading="lazy"></dd></dl>

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.<a href="#cite_note-16">[16]</a>

In the specific case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle z=w,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle z=w,}</annotation>

</semantics>

</math><img src="./1e56d9cd9e21a79d8e67571975e3b2e6b135ff63.svg" 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,}" loading="lazy"> double-argument identities are

<mtr>

<mtd>

</mtd>

<mtd>

<mi>arctan</mi>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<mi>artanh</mi>

<msup>

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</msup>

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<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><img src="./72a7f2451e5d3d5827bc599e1c3d711795d73c24.svg" 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}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Taylor_series">Taylor series</h2></div>

The <a href="Taylor_series" title="Taylor series">Taylor series</a> near zero, valid for complex values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle z}</annotation>

</semantics>

</math><img src="./7a33e37010e3acdeeb80fdb95df9bfe411fd79e6.svg" 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}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |z|<{\tfrac {1}{2}}\pi ,}">

</mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle |z|<{\tfrac {1}{2}}\pi ,}</annotation>

</semantics>

</math><img src="./766bcb8ac76422ca5a8466525dd459757489ef8c.svg" 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 ,}" loading="lazy"> are<a href="#cite_note-17">[17]</a>

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</munderover>

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</munderover>

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</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><img src="./c867f2c3e83d3142bc5de97d045c1311b0d30203.svg" 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}}}" loading="lazy"></dd></dl>

where the numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle E_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle E_{k}}</annotation>

</semantics>

</math><img src="./62abd82e78906093d1d81a06bc46e926c35ab0fe.svg" 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}}" loading="lazy"> are the <a href="Euler_numbers" title="Euler numbers">Euler secant numbers</a>, 1, 0, -1, 0, 5, 0, -61, 0, 1385 ... (sequences <a href="https://oeis.org/A122045" class="extiw external" title="oeis:A122045">A122045</a>, <a href="https://oeis.org/A000364" class="extiw external" title="oeis:A000364">A000364</a>, and <a href="https://oeis.org/A028296" class="extiw external" title="oeis:A028296">A028296</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). These series were first computed by <a href="James_Gregory_(mathematician)" title="James Gregory (mathematician)">James Gregory</a> in 1671.<a href="#cite_note-18">[18]</a>

Because the Gudermannian and inverse Gudermannian functions are the integrals of the hyperbolic secant and secant functions, the numerators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle E_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle E_{k}}</annotation>

</semantics>

</math><img src="./62abd82e78906093d1d81a06bc46e926c35ab0fe.svg" 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}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |E_{k}|}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle |E_{k}|}</annotation>

</semantics>

</math><img src="./313e5e861810b966807e07241b67a9416e0c1345.svg" 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}|}" loading="lazy"> are same as the numerators of the <a href="Hyperbolic_functions#Taylor_series_expressions" title="Hyperbolic functions">Taylor series for sech</a> and <a href="Trigonometric_functions#Power_series_expansion" title="Trigonometric functions">sec</a>, respectively, but shifted by one place.

The reduced unsigned numerators are 1, 1, 1, 61, 277, ... and the reduced denominators are 1, 6, 24, 5040, 72576, ... (sequences <a href="https://oeis.org/A091912" class="extiw external" title="oeis:A091912">A091912</a> and <a href="https://oeis.org/A136606" class="extiw external" title="oeis:A136606">A136606</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

/* start https://en.wikipedia.org/ */

.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}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">For broader coverage of this topic, see <a href="Mercator_projection#History" title="Mercator projection">Mercator projection § History</a>, and <a href="Integral_of_the_secant_function" title="Integral of the secant function">Integral of the secant function</a>.</div>

The function and its inverse are related to the <a href="Mercator_projection" title="Mercator projection">Mercator projection</a>. The vertical coordinate in the Mercator projection is called <a href="Latitude#Isometric_latitude" title="Latitude">isometric latitude</a>, and is often denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \psi .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \psi .}</annotation>

</semantics>

</math><img src="./0f441ae5145c06b5b873eef121d91d3efdd75f35.svg" 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 .}" loading="lazy"> In terms of <a href="Latitude" title="Latitude">latitude</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \phi }</annotation>

</semantics>

</math><img src="./b36dc4887c86d6f01183292a5d2f984e23318c10.svg" 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 }" loading="lazy"> on the sphere (expressed in <a href="Radian" title="Radian">radians</a>) the isometric latitude can be written

<msup>

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</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi =\operatorname {gd} ^{-1}\phi =\int _{0}^{\phi }\sec t\,\mathrm {d} t.}</annotation>

</semantics>

</math><img src="./a920098bfd9d3754f6573874769b15d15710c40e.svg" 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.}" loading="lazy"></dd></dl>

The inverse from the isometric latitude to spherical latitude is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi =\operatorname {gd} \psi .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \phi =\operatorname {gd} \psi .}</annotation>

</semantics>

</math><img src="./4094fcc850b702061e791c0637d9d53f0a7e5608.svg" 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 .}" loading="lazy"> (Note: on an <a href="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.)

<a href="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="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="James_Gregory_(mathematician)" title="James Gregory (mathematician)">James Gregory</a>.

The Gudermannian function per se was introduced by <a href="Johann_Heinrich_Lambert" title="Johann Heinrich Lambert">Johann Heinrich Lambert</a> in the 1760s at the same time as the <a href="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="Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a> suggested it be given its current name as a tribute to <a href="Christoph_Gudermann" title="Christoph Gudermann">Christoph Gudermann</a>'s work in the 1830s on the theory of special functions.<a href="#cite_note-19">[19]</a>

Gudermann had published articles in <a href="Crelle's_Journal" title="Crelle's Journal">Crelle's Journal</a> that were later collected in a book<a href="#cite_note-20">[20]</a>

which expounded <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sinh }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \sinh }</annotation>

</semantics>

</math><img src="./f54dc39a77cadfbe93dff7a9ab383194091c2513.svg" 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 }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \cosh }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \cosh }</annotation>

</semantics>

</math><img src="./3419abb2bed522294ca025fb5ffd9fff9d7d0bdb.svg" 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 }" loading="lazy"> to a wide audience (although represented by the symbols <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathfrak {Sin}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\mathfrak {Sin}}}</annotation>

</semantics>

</math><img src="./ecefb7a444fdbe6d637755859e58195c9995e293.svg" 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}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathfrak {Cos}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\mathfrak {Cos}}}</annotation>

</semantics>

</math><img src="./b5c823e02b6f2d73871c88e697d261f802867cc3.svg" 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}}}" loading="lazy">).

The notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {gd} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {gd} }</annotation>

</semantics>

</math><img src="./9e8496c6ca1328821c3f42dd1173a63d6dacbd05.svg" 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} }" loading="lazy"> was introduced by Cayley who starts by calling <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi =\operatorname {gd} u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \phi =\operatorname {gd} u}</annotation>

</semantics>

</math><img src="./ed00c253b5012aa6edefa172488beb2c8ed663c9.svg" 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}" loading="lazy"> the <a href="Jacobi_elliptic_functions#am" title="Jacobi elliptic functions">Jacobi elliptic amplitude</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {am} u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {am} u}</annotation>

</semantics>

</math><img src="./3c8109fe0f7e018c9f4c8b61cd0ee683a4a1efd4.svg" 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}" loading="lazy"> in the degenerate case where the elliptic modulus is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle m=1,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle m=1,}</annotation>

</semantics>

</math><img src="./90c13cd3deaf3bd232b483e7a04df890d0cdd84c.svg" 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,}" loading="lazy"> so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {1-m\sin \!^{2}\,\phi }}}">

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\sqrt {1-m\sin \!^{2}\,\phi }}}</annotation>

</semantics>

</math><img src="./d055d17634713cdac511fd75c6a084f68bb8fa1c.svg" 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 }}}" loading="lazy"> reduces to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \cos \phi .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \cos \phi .}</annotation>

</semantics>

</math><img src="./dbe802eb64df25a701e15ba2b699ef62981470b2.svg" 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 .}" loading="lazy"><a href="#cite_note-21">[21]</a> This is the inverse of the <a href="Integral_of_the_secant_function" title="Integral of the secant function">integral of the secant function</a>. Using Cayley's notation,

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</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><img src="./b8fc69dc7a8fcb5c4f28ead6392400cabe2041c8.svg" 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 )}.}" loading="lazy"></dd></dl>

He then derives "the definition of the transcendent",

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</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><img src="./3e97e04d4db5470015af2754956b88596bc806cd.svg" 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 )},}" loading="lazy"></dd></dl>

observing that "although exhibited in an imaginary form, [it] is a real function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle u}</annotation>

</semantics>

</math><img src="./24e12e26e505ed5b02c7648a89bbc6737038b2de.svg" 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}" loading="lazy">".

The Gudermannian and its inverse were used to make <a href="Trigonometric_tables" title="Trigonometric tables">trigonometric tables</a> of circular functions also function as tables of hyperbolic functions. Given a hyperbolic angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \psi }</annotation>

</semantics>

</math><img src="./9d738e3571903ec4e786923ddbd817cd147cb5a2.svg" 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 }" loading="lazy">, hyperbolic functions could be found by first looking up <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi =\operatorname {gd} \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \phi =\operatorname {gd} \psi }</annotation>

</semantics>

</math><img src="./a90392c8dc8bdc35d51f9b99dd6d134d4300a4b0.svg" 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 }" loading="lazy"> in a Gudermannian table and then looking up the appropriate circular function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \phi }</annotation>

</semantics>

</math><img src="./b36dc4887c86d6f01183292a5d2f984e23318c10.svg" 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 }" loading="lazy">, or by directly locating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \psi }</annotation>

</semantics>

</math><img src="./9d738e3571903ec4e786923ddbd817cd147cb5a2.svg" 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 }" loading="lazy"> in an auxiliary <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gd} ^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {gd} ^{-1}}</annotation>

</semantics>

</math><img src="./1a79e1a6ee5398ea160393c746ff96902f6857fa.svg" 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}}" loading="lazy"> column of the trigonometric table.<a href="#cite_note-22">[22]</a>

<div class="mw-heading mw-heading2"><h2 id="Generalization">Generalization</h2></div>

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 n-dimensional <a href="Hyperboloid" title="Hyperboloid">hyperboloid of two sheets</a> can be likewise mapped onto a n-dimensional hemisphere via stereographic projection. The <a href="Hyperbolic_geometry#The_hemisphere_model" title="Hyperbolic geometry">hemisphere model of hyperbolic space</a> uses such a map to represent hyperbolic space.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<ul><li>The <a href="Angle_of_parallelism" title="Angle of parallelism">angle of parallelism</a> function in <a href="Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a> is the <a href="Angle#Combining_angle_pairs" title="Angle">complement</a> of the gudermannian, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {\Pi }}(\psi )={\tfrac {1}{2}}\pi -\operatorname {gd} \psi .}">

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathit {\Pi }}(\psi )={\tfrac {1}{2}}\pi -\operatorname {gd} \psi .}</annotation>

</semantics>

</math><img src="./d69561be2de197f573d58ce3bc54e48999d5f625.svg" 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 .}" loading="lazy"></li>

<li>On a <a href="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="Transverse_Mercator_projection" title="Transverse Mercator projection">transverse Mercator projection</a>.<a href="#cite_note-23">[23]</a></li>

<li>The Gudermannian function appears in a non-periodic solution of the <a href="Inverted_pendulum" title="Inverted pendulum">inverted pendulum</a>.<a href="#cite_note-24">[24]</a></li>

<li>The Gudermannian function appears in a moving mirror solution of the dynamical <a href="Casimir_effect" title="Casimir effect">Casimir effect</a>.<a href="#cite_note-25">[25]</a></li>

<li>If an infinite number of infinitely long, equidistant, parallel, coplanar, straight wires are kept at equal <a href="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.<a href="#cite_note-26">[26]</a></li>

<li>The Gudermannian function is a <a href="Sigmoid_function" title="Sigmoid function">sigmoid function</a>, and as such is sometimes used as an <a href="Activation_function" title="Activation function">activation function</a> in machine learning.</li>

<li>The (scaled and shifted) Gudermannian function is the <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of the <a href="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="Spiral_galaxy" title="Spiral galaxy">spiral galaxy</a> arms.<a href="#cite_note-27">[27]</a></li></ul>

<ul><li><a href="Tractrix" title="Tractrix">Tractrix</a></li>

<li><a href="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></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> The symbols <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \psi }</annotation>

</semantics>

</math><img src="./9d738e3571903ec4e786923ddbd817cd147cb5a2.svg" 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 }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \phi }</annotation>

</semantics>

</math><img src="./b36dc4887c86d6f01183292a5d2f984e23318c10.svg" 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 }" loading="lazy"> were chosen for this article because they are commonly used in <a href="Geodesy" title="Geodesy">geodesy</a> for the <a href="Latitude#Isometric_latitude" title="Latitude">isometric latitude</a> (vertical coordinate of the <a href="Mercator_projection" title="Mercator projection">Mercator projection</a>) and <a href="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.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Gudermann published several papers about the trigonometric and hyperbolic functions in <a href="Crelle's_Journal" title="Crelle's Journal">Crelle's Journal</a> in 1830–1831. These were collected in a book, <a href="#CITEREFGudermann1833">Gudermann (1833)</a>.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <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>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFKennelly1929">Kennelly (1929)</a>; <a href="#CITEREFLee1976">Lee (1976)</a>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFMasson2021">Masson (2021)</a>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFGottschalk2003">Gottschalk (2003)</a> pp. 23–27

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <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.

</li>

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</style><cite id="CITEREFWeisstein" class="citation web cs1"><a href="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>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#CITEREFKennelly1929">Kennelly (1929)</a>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <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>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</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> – note the typo.

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <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>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <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>

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <a href="#CITEREFKahligReich2013">Kahlig & Reich (2013)</a>

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <a href="#CITEREFCayley1862">Cayley (1862)</a> <a rel="nofollow" class="external text" href="https://archive.org/details/londonedinburg4241862lond/page/21">p. 21</a>

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <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>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <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>

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFTurnbull1939" class="citation book cs1">Turnbull, Herbert Westren, ed. (1939). James Gregory; Tercentenary Memorial Volume. G. Bell & Sons. p. 170.</cite>

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <a href="#CITEREFBeckerVan_Orstrand1909">Becker & Van Orstrand (1909)</a>

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> <a href="#CITEREFGudermann1833">Gudermann (1833)</a>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <a href="#CITEREFCayley1862">Cayley (1862)</a>

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> For example Hoüel labels the hyperbolic functions across the top in Table XIV of: <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/">Recueil de formules et de tables numériques</a>. Gauthier-Villars. p. 36.</cite>

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <a href="#CITEREFOsborne2013">Osborne (2013)</a> p. 74

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <a href="#CITEREFRobertson1997">Robertson (1997)</a>

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> <a href="#CITEREFGoodAndersonEvans2013">Good, Anderson & Evans (2013)</a>

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> <a href="#CITEREFKennelly1928">Kennelly (1928)</a>

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> <a href="#CITEREFRingermacherMead2009">Ringermacher & Mead (2009)</a>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<annotation encoding="application/x-tex">{\textstyle \operatorname {gd} u={\tfrac {1}{i}}\log \tan {\bigl (}{\tfrac {1}{4}}\pi +{\tfrac {1}{2}}ui{\bigr )}}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li>Penn, Michael (2020) <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=rypoQgdF5cM">"the Gudermannian function!"</a> on YouTube.</li></ul></div><div><div class="zim-footer">

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