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Gudermannian function

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(Redirected fromTranscendent angle)

Mathematical function relating circular and hyperbolic functions

The Gudermannian function relates the area of acircular sector to the area of ahyperbolic sector, via a commonstereographic projection. If twice the area of the blue hyperbolic sector isψ, then twice the area of the red circular sector isϕ = gdψ. Twice the area of the purple triangle is the stereographic projections = tan⁠1/2⁠ϕ = tanh⁠1/2⁠ψ. The blue point has coordinates(coshψ, sinhψ). The red point has coordinates(cosϕ, sinϕ). The purple point has coordinates(0,s).

Graph of the inverse Gudermannian function.

In mathematics, theGudermannian function relates ahyperbolic angle measure ${\textstyle \psi }$ to acircular angle measure ${\textstyle \phi }$ called thegudermannian of ${\textstyle \psi }$ and denoted ${\textstyle \operatorname {gd} \psi }$ .^[1] The Gudermannian function reveals a close relationship between thecircular functions andhyperbolic functions. It was introduced in the 1760s byJohann Heinrich Lambert, and later named forChristoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830.^[2] The gudermannian is sometimes called thehyperbolic amplitude as alimiting case of theJacobi elliptic amplitude ${\textstyle \operatorname {am} (\psi ,m)}$ when parameter ${\textstyle m=1.}$

Thereal Gudermannian function is typically defined for ${\textstyle -\infty <\psi <\infty }$ to be the integral of the hyperbolic secant^[3]

\phi =\operatorname {gd} \psi \equiv \int _{0}^{\psi }\operatorname {sech} t\,\mathrm {d} t=\operatorname {arctan} (\sinh \psi ).

The real inverse Gudermannian function can be defined for ${\textstyle -{\tfrac {1}{2}}\pi <\phi <{\tfrac {1}{2}}\pi }$ as theintegral of the (circular) secant

\psi =\operatorname {gd} ^{-1}\phi =\int _{0}^{\phi }\operatorname {sec} t\,\mathrm {d} t=\operatorname {arsinh} (\tan \phi ).

The hyperbolic angle measure $\psi =\operatorname {gd} ^{-1}\phi$ is called theanti-gudermannian of $\phi$ or sometimes thelambertian of $\phi$ , denoted $\psi =\operatorname {lam} \phi .$ ^[4] In the context ofgeodesy andnavigation for latitude ${\textstyle \phi }$ , $k\operatorname {gd} ^{-1}\phi$ (scaled by arbitrary constant ${\textstyle k}$ ) was historically called themeridional part of $\phi$ (French:latitude croissante). It is the vertical coordinate of theMercator projection.

The two angle measures ${\textstyle \phi }$ and ${\textstyle \psi }$ are related by a commonstereographic projection

s=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi ,

and this identity can serve as an alternative definition for ${\textstyle \operatorname {gd} }$ and ${\textstyle \operatorname {gd} ^{-1}}$ valid throughout thecomplex plane:

{\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}}

Circular–hyperbolic identities

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We can evaluate the integral of the hyperbolic secant using the stereographic projection (hyperbolic half-tangent) as achange of variables:^[5]

{\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}}

Letting ${\textstyle \phi =\operatorname {gd} \psi }$ and ${\textstyle s=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi }$ we can derive a number of identities between hyperbolic functions of ${\textstyle \psi }$ and circular functions of ${\textstyle \phi .}$ ^[6]

Identities related to the Gudermannian function represented graphically.

{\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}}

These are commonly used as expressions for $\operatorname {gd}$ and $\operatorname {gd} ^{-1}$ for real values of $\psi$ and $\phi$ with $|\phi |<{\tfrac {1}{2}}\pi .$ For example, the numerically well-behaved formulas

{\begin{aligned}\operatorname {gd} \psi &=\operatorname {arctan} (\sinh \psi ),\\[6mu]\operatorname {gd} ^{-1}\phi &=\operatorname {arsinh} (\tan \phi ).\end{aligned}}

(Note, for $|\phi |>{\tfrac {1}{2}}\pi$ and for complex arguments, care must be taken choosingbranches of the inverse functions.)^[7]

We can also express ${\textstyle \psi }$ and ${\textstyle \phi }$ in terms of ${\textstyle s\colon }$

{\begin{aligned}2\arctan s&=\phi =\operatorname {gd} \psi ,\\[6mu]2\operatorname {artanh} s&=\operatorname {gd} ^{-1}\phi =\psi .\\[6mu]\end{aligned}}

If we expand ${\textstyle \tan {\tfrac {1}{2}}}$ and ${\textstyle \tanh {\tfrac {1}{2}}}$ in terms of theexponential, then we can see that ${\textstyle s,}$ $\exp \phi i,$ and $\exp \psi$ are allMöbius transformations of each-other (specifically, rotations of theRiemann sphere):

{\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}}

For real values of ${\textstyle \psi }$ and ${\textstyle \phi }$ with $|\phi |<{\tfrac {1}{2}}\pi$ , these Möbius transformations can be written in terms of trigonometric functions in several ways,

{\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}}

These give further expressions for $\operatorname {gd}$ and $\operatorname {gd} ^{-1}$ for real arguments with $|\phi |<{\tfrac {1}{2}}\pi .$ For example,^[8]

{\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}}

Complex values

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The Gudermannian functionz ↦ gdz is a conformal map from an infinite strip to an infinite strip. It can be broken into two parts: a mapz ↦ tanh⁠1/2⁠z from one infinite strip to the complex unit disk and a mapζ ↦ 2 arctanζ from the disk to the other infinite strip.

As afunction of a complex variable, ${\textstyle z\mapsto w=\operatorname {gd} z}$ conformally maps the infinite strip ${\textstyle \left|\operatorname {Im} z\right|\leq {\tfrac {1}{2}}\pi }$ to the infinite strip ${\textstyle \left|\operatorname {Re} w\right|\leq {\tfrac {1}{2}}\pi ,}$ while ${\textstyle w\mapsto z=\operatorname {gd} ^{-1}w}$ conformally maps the infinite strip ${\textstyle \left|\operatorname {Re} w\right|\leq {\tfrac {1}{2}}\pi }$ to the infinite strip ${\textstyle \left|\operatorname {Im} z\right|\leq {\tfrac {1}{2}}\pi .}$

Analytically continued byreflections to the whole complex plane, ${\textstyle z\mapsto w=\operatorname {gd} z}$ is a periodic function of period ${\textstyle 2\pi i}$ which sends any infinite strip of "height" ${\textstyle 2\pi i}$ onto the strip ${\textstyle -\pi <\operatorname {Re} w\leq \pi .}$ Likewise, extended to the whole complex plane, ${\textstyle w\mapsto z=\operatorname {gd} ^{-1}w}$ is a periodic function of period ${\textstyle 2\pi }$ which sends any infinite strip of "width" ${\textstyle 2\pi }$ onto the strip ${\textstyle -\pi <\operatorname {Im} z\leq \pi .}$ ^[9] For all points in the complex plane, these functions can be correctly written as:

{\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}}

For the ${\textstyle \operatorname {gd} }$ and ${\textstyle \operatorname {gd} ^{-1}}$ functions to remain invertible with these extended domains, we might consider each to be amultivalued function (perhaps ${\textstyle \operatorname {Gd} }$ and ${\textstyle \operatorname {Gd} ^{-1}}$ , with ${\textstyle \operatorname {gd} }$ and ${\textstyle \operatorname {gd} ^{-1}}$ theprincipal branch) or consider their domains and codomains asRiemann surfaces.

If ${\textstyle u+iv=\operatorname {gd} (x+iy),}$ then the real and imaginary components ${\textstyle u}$ and ${\textstyle v}$ can be found by:^[10]

\tan u={\frac {\sinh x}{\cos y}},\quad \tanh v={\frac {\sin y}{\cosh x}}.

(In practical implementation, make sure to use the2-argument arctangent, ${\textstyle u=\operatorname {atan2} (\sinh x,\cos y)}$ .)

Likewise, if ${\textstyle x+iy=\operatorname {gd} ^{-1}(u+iv),}$ then components ${\textstyle x}$ and ${\textstyle y}$ can be found by:^[11]

\tanh x={\frac {\sin u}{\cosh v}},\quad \tan y={\frac {\sinh v}{\cos u}}.

Multiplying these together reveals the additional identity^[8]

\tanh x\,\tan y=\tan u\,\tanh v.

Symmetries

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The two functions can be thought of as rotations or reflections of each-other, with a similar relationship as ${\textstyle \sinh iz=i\sin z}$ between sine and hyperbolic sine:^[12]

{\begin{aligned}\operatorname {gd} iz&=i\operatorname {gd} ^{-1}z,\\[5mu]\operatorname {gd} ^{-1}iz&=i\operatorname {gd} z.\end{aligned}}

The functions are bothodd and they commute withcomplex conjugation. That is, a reflection across the real or imaginary axis in the domain results in the same reflection in thecodomain:

{\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}}

The functions areperiodic, with periods ${\textstyle 2\pi i}$ and ${\textstyle 2\pi }$ :

{\begin{aligned}\operatorname {gd} (z+2\pi i)&=\operatorname {gd} z,\\[5mu]\operatorname {gd} ^{-1}(z+2\pi )&=\operatorname {gd} ^{-1}z.\end{aligned}}

A translation in the domain of ${\textstyle \operatorname {gd} }$ by ${\textstyle \pm \pi i}$ results in a half-turn rotation and translation in the codomain by one of ${\textstyle \pm \pi ,}$ and vice versa for ${\textstyle \operatorname {gd} ^{-1}\colon }$ ^[13]

{\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}}

A reflection in the domain of ${\textstyle \operatorname {gd} }$ across either of the lines ${\textstyle x\pm {\tfrac {1}{2}}\pi i}$ results in a reflection in the codomain across one of the lines ${\textstyle \pm {\tfrac {1}{2}}\pi +yi,}$ and vice versa for ${\textstyle \operatorname {gd} ^{-1}\colon }$

{\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}}

This is related to the identity

\tanh {\tfrac {1}{2}}({\pi i}\pm z)=\tan {\tfrac {1}{2}}({\pi }\mp \operatorname {gd} z).

Specific values

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A few specific values (where ${\textstyle \infty }$ indicates the limit at one end of the infinite strip):^[14]

{\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}}

Derivatives

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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:

{\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}}

Argument-addition identities

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By combininghyperbolic andcircular argument-addition identities,

{\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}}

with thecircular–hyperbolic identity,

\tan {\tfrac {1}{2}}(\operatorname {gd} z)=\tanh {\tfrac {1}{2}}z,

we have the Gudermannian argument-addition identities:

{\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}}

Further argument-addition identities can be written in terms of other circular functions,^[15] but they require greater care in choosing branches in inverse functions. Notably,

{\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}}

which can be used to derive theper-component computation for the complex Gudermannian and inverse Gudermannian.^[16]

In the specific case ${\textstyle z=w,}$ double-argument identities are

{\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}}

Taylor series

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TheTaylor series near zero, valid for complex values ${\textstyle z}$ with ${\textstyle |z|<{\tfrac {1}{2}}\pi ,}$ are^[17]

{\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}}

where the numbers ${\textstyle E_{k}}$ are theEuler secant numbers, 1, 0, -1, 0, 5, 0, -61, 0, 1385 ... (sequencesA122045,A000364, andA028296 in theOEIS). These series were first computed byJames Gregory in 1671.^[18]

Because the Gudermannian and inverse Gudermannian functions are the integrals of the hyperbolic secant and secant functions, the numerators ${\textstyle E_{k}}$ and ${\textstyle |E_{k}|}$ are same as the numerators of theTaylor series forsech andsec, 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, ... (sequencesA091912 andA136606 in theOEIS).

History

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For broader coverage of this topic, seeMercator projection § History, andIntegral of the secant function.

The function and its inverse are related to theMercator projection. The vertical coordinate in the Mercator projection is calledisometric latitude, and is often denoted ${\textstyle \psi .}$ In terms oflatitude ${\textstyle \phi }$ on the sphere (expressed inradians) the isometric latitude can be written

\psi =\operatorname {gd} ^{-1}\phi =\int _{0}^{\phi }\sec t\,\mathrm {d} t.

The inverse from the isometric latitude to spherical latitude is ${\textstyle \phi =\operatorname {gd} \psi .}$ (Note: on anellipsoid of revolution, the relation between geodetic latitude and isometric latitude is slightly more complicated.)

Gerardus Mercator plotted his celebrated map in 1569, but the precise method of construction was not revealed. In 1599,Edward Wright 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 byJames Gregory.

The Gudermannian function per se was introduced byJohann Heinrich Lambert in the 1760s at the same time as thehyperbolic functions. He called it the "transcendent angle", and it went by various names until 1862 whenArthur Cayley suggested it be given its current name as a tribute toChristoph Gudermann's work in the 1830s on the theory ofspecial functions.^[19]Gudermann had published articles inCrelle's Journal that were later collected in a book^[20]which expounded ${\textstyle \sinh }$ and ${\textstyle \cosh }$ to a wide audience (although represented by the symbols ${\textstyle {\mathfrak {Sin}}}$ and ${\textstyle {\mathfrak {Cos}}}$ ).

The notation ${\textstyle \operatorname {gd} }$ was introduced by Cayley who starts by calling ${\textstyle \phi =\operatorname {gd} u}$ theJacobi elliptic amplitude ${\textstyle \operatorname {am} u}$ in the degenerate case where the elliptic modulus is ${\textstyle m=1,}$ so that ${\textstyle {\sqrt {1-m\sin \!^{2}\,\phi }}}$ reduces to ${\textstyle \cos \phi .}$ ^[21] This is the inverse of theintegral of the secant function. Using Cayley's notation,

u=\int _{0}{\frac {d\phi }{\cos \phi }}={\log \,\tan }{\bigl (}{\tfrac {1}{4}}\pi +{\tfrac {1}{2}}\phi {\bigr )}.

He then derives "the definition of the transcendent",

\operatorname {gd} u={{\frac {1}{i}}\log \,\tan }{\bigl (}{\tfrac {1}{4}}\pi +{\tfrac {1}{2}}ui{\bigr )},

observing that "although exhibited in an imaginary form, [it] is a real function of ${\textstyle u}$ ".

The Gudermannian and its inverse were used to maketrigonometric tables of circular functions also function as tables of hyperbolic functions. Given a hyperbolic angle ${\textstyle \psi }$ , hyperbolic functions could be found by first looking up ${\textstyle \phi =\operatorname {gd} \psi }$ in a Gudermannian table and then looking up the appropriate circular function of ${\textstyle \phi }$ , or by directly locating ${\textstyle \psi }$ in an auxiliary $\operatorname {gd} ^{-1}$ column of the trigonometric table.^[22]

Generalization

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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 ann-dimensionalhyperboloid of two sheets can be likewise mapped onto an-dimensional hemisphere via stereographic projection. Thehemisphere model of hyperbolic space uses such a map to represent hyperbolic space.

Applications

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Distance in thePoincaré half-plane model of thehyperbolic plane from the apex of a semicircle to another point on it is the inverse Gudermannian function of the central angle.

Theangle of parallelism function inhyperbolic geometry is thecomplement of the gudermannian, ${\mathit {\Pi }}(\psi )={\tfrac {1}{2}}\pi -\operatorname {gd} \psi .$
On aMercator projection a line of constant latitude is parallel to the equator (on the projection) at a distance proportional to the anti-gudermannian of the latitude.
The Gudermannian function (with acomplex argument) may be used to define thetransverse Mercator projection.^[23]
The Gudermannian function appears in a non-periodic solution of theinverted pendulum.^[24]
The Gudermannian function appears in a moving mirror solution of the dynamicalCasimir effect.^[25]
If an infinite number of infinitely long, equidistant, parallel, coplanar, straight wires are kept at equalpotentials with alternating signs, the potential-flux distribution in a cross-sectional plane perpendicular to the wires is the complex Gudermannian function.^[26]
The Gudermannian function is asigmoid function, and as such is sometimes used as anactivation function in machine learning.
The (scaled and shifted) Gudermannian function is thecumulative distribution function of thehyperbolic secant distribution.
A function based on the Gudermannian provides a good model for the shape ofspiral galaxy arms.^[27]