Gudermannian function

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

${\displaystyle {\begin{aligned}\operatorname {gd} \psi &\equiv \int _{0}^{\psi }{\frac {1}{\operatorname {cosh} t}}\mathrm {d} t=\int _{0}^{\tanh {\frac {1}{2}}\psi }{\frac {1-u^{2}}{1+u^{2}}}{\frac {2\,\mathrm {d} u}{1-u^{2}}}\qquad {\bigl (}u=\tanh {\tfrac {1}{2}}t{\bigr )}\\[8mu]&=2\int _{0}^{\tanh {\frac {1}{2}}\psi }{\frac {1}{1+u^{2}}}\mathrm {d} u={2\arctan }{\bigl (}\tanh {\tfrac {1}{2}}\psi \,{\bigr )},\\[5mu]\tan {\tfrac {1}{2}}{\operatorname {gd} \psi }&=\tanh {\tfrac {1}{2}}\psi .\end{aligned}}}$

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.

${\displaystyle {\begin{aligned}s&=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi ,\\[6mu]{\frac {2s}{1+s^{2}}}&=\sin \phi =\tanh \psi ,\quad &{\frac {1+s^{2}}{2s}}&=\csc \phi =\coth \psi ,\\[10mu]{\frac {1-s^{2}}{1+s^{2}}}&=\cos \phi =\operatorname {sech} \psi ,\quad &{\frac {1+s^{2}}{1-s^{2}}}&=\sec \phi =\cosh \psi ,\\[10mu]{\frac {2s}{1-s^{2}}}&=\tan \phi =\sinh \psi ,\quad &{\frac {1-s^{2}}{2s}}&=\cot \phi =\operatorname {csch} \psi .\\[8mu]\end{aligned}}}$

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

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

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

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

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

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

${\displaystyle {\begin{aligned}s&=i{\frac {1-e^{\phi i}}{1+e^{\phi i}}}={\frac {e^{\psi }-1}{e^{\psi }+1}},\\[10mu]i{\frac {s-i}{s+i}}&=\exp \phi i\quad ={\frac {e^{\psi }-i}{e^{\psi }+i}},\\[10mu]{\frac {1+s}{1-s}}&=i{\frac {i+e^{\phi i}}{i-e^{\phi i}}}\,=\exp \psi .\end{aligned}}}$

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

${\displaystyle {\begin{aligned}\exp \psi &=\sec \phi +\tan \phi =\tan {\tfrac {1}{2}}{\bigl (}{\tfrac {1}{2}}\pi +\phi {\bigr )}\\[6mu]&={\frac {1+\tan {\tfrac {1}{2}}\phi }{1-\tan {\tfrac {1}{2}}\phi }}={\sqrt {\frac {1+\sin \phi }{1-\sin \phi }}},\\[12mu]\exp \phi i&=\operatorname {sech} \psi +i\tanh \psi =\tanh {\tfrac {1}{2}}{\bigl (}{-{\tfrac {1}{2}}}\pi i+\psi {\bigr )}\\[6mu]&={\frac {1+i\tanh {\tfrac {1}{2}}\psi }{1-i\tanh {\tfrac {1}{2}}\psi }}={\sqrt {\frac {1+i\sinh \psi }{1-i\sinh \psi }}}.\end{aligned}}}$

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

${\displaystyle {\begin{aligned}\operatorname {gd} \psi &=2\arctan e^{\psi }-{\tfrac {1}{2}}\pi ,\\[6mu]\operatorname {gd} ^{-1}\phi &=\log(\sec \phi +\tan \phi ).\end{aligned}}}$

The Gudermannian function z ↦ gd z is a conformal map from an infinite strip to an infinite strip. It can be broken into two parts: a map z ↦ tanh 1/2z from one infinite strip to the complex unit disk and a map ζ ↦ 2 arctan ζ from the disk to the other infinite strip.

As a functions 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 by reflections 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:

${\displaystyle {\begin{aligned}\operatorname {gd} z&={2\arctan }{\bigl (}\tanh {\tfrac {1}{2}}z\,{\bigr )},\\[5mu]\operatorname {gd} ^{-1}w&={2\operatorname {artanh} }{\bigl (}\tan {\tfrac {1}{2}}w\,{\bigr )}.\end{aligned}}}$

For the ${\textstyle \operatorname {gd} }$ and ${\textstyle \operatorname {gd} ^{-1}}$ functions to remain invertible with these extended domains, we might consider each to be a multivalued function (perhaps ${\textstyle \operatorname {Gd} }$ and ${\textstyle \operatorname {Gd} ^{-1}}$, with ${\textstyle \operatorname {gd} }$ and ${\textstyle \operatorname {gd} ^{-1}}$ the principal branch) or consider their domains and codomains as Riemann 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]

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

(In practical implementation, make sure to use the 2-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]

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

Multiplying these together reveals the additional identity[8]

${\displaystyle \tanh x\,\tan y=\tan u\,\tanh v.}$

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {gd} z&=\operatorname {sech} z,\\[10mu]{\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {gd} ^{-1}z&=\sec z.\end{aligned}}}$

By combining hyperbolic and circular argument-addition identities,

${\displaystyle {\begin{aligned}\tanh(z+w)&={\frac {\tanh z+\tanh w}{1+\tanh z\,\tanh w}},\\[10mu]\tan(z+w)&={\frac {\tan z+\tan w}{1-\tan z\,\tan w}},\end{aligned}}}$

with the circular–hyperbolic identity,

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

we have the Gudermannian argument-addition identities:

${\displaystyle {\begin{aligned}\operatorname {gd} (z+w)&=2\arctan {\frac {\tan {\tfrac {1}{2}}(\operatorname {gd} z)+\tan {\tfrac {1}{2}}(\operatorname {gd} w)}{1+\tan {\tfrac {1}{2}}(\operatorname {gd} z)\,\tan {\tfrac {1}{2}}(\operatorname {gd} w)}},\\[12mu]\operatorname {gd} ^{-1}(z+w)&=2\operatorname {artanh} {\frac {\tanh {\tfrac {1}{2}}(\operatorname {gd} ^{-1}z)+\tanh {\tfrac {1}{2}}(\operatorname {gd} ^{-1}w)}{1-\tanh {\tfrac {1}{2}}(\operatorname {gd} ^{-1}z)\,\tanh {\tfrac {1}{2}}(\operatorname {gd} ^{-1}w)}}.\end{aligned}}}$

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,

${\displaystyle {\begin{aligned}\operatorname {gd} (z+w)&=u+v,\quad {\text{where}}\ \tan u={\frac {\sinh z}{\cosh w}},\ \tan v={\frac {\sinh w}{\cosh z}},\\[10mu]\operatorname {gd} ^{-1}(z+w)&=u+v,\quad {\text{where}}\ \tanh u={\frac {\sin z}{\cos w}},\ \tanh v={\frac {\sin w}{\cos z}},\end{aligned}}}$

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

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

${\displaystyle {\begin{aligned}\operatorname {gd} (2z)&=2\arctan(\sin(\operatorname {gd} z)),\\[5mu]\operatorname {gd} ^{-1}(2z)&=2\operatorname {artanh} (\sinh(\operatorname {gd} ^{-1}z)).\end{aligned}}}$

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

${\displaystyle {\begin{aligned}\operatorname {gd} z&=\sum _{k=0}^{\infty }{\frac {E_{k}}{(k+1)!}}z^{k+1}=z-{\frac {1}{6}}z^{3}+{\frac {1}{24}}z^{5}-{\frac {61}{5040}}z^{7}+{\frac {277}{72576}}z^{9}-\dots ,\\[10mu]\operatorname {gd} ^{-1}z&=\sum _{k=0}^{\infty }{\frac {|E_{k}|}{(k+1)!}}z^{k+1}=z+{\frac {1}{6}}z^{3}+{\frac {1}{24}}z^{5}+{\frac {61}{5040}}z^{7}+{\frac {277}{72576}}z^{9}+\dots ,\end{aligned}}}$

where the numbers ${\textstyle E_{k}}$ are the Euler secant numbers, 1, 0, -1, 0, 5, 0, -61, 0, 1385 ... (sequences A122045, A000364, and A028296 in the OEIS). These series were first computed by James 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 the Taylor series for sech and sec, 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 A091912 and A136606 in the OEIS).

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

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

The inverse from the isometric latitude to spherical latitude is ${\textstyle \phi =\operatorname {gd} \psi .}$ (Note: on an ellipsoid 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 by James Gregory.

The Gudermannian function per se was introduced by Johann Heinrich Lambert in the 1760s at the same time as the hyperbolic functions. He called it the "transcendent angle", and it went by various names until 1862 when Arthur Cayley suggested it be given its current name as a tribute to Christoph Gudermann's work in the 1830s on the theory of special functions.[19] Gudermann had published articles in Crelle'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 ${\textstyle \phi =\operatorname {gd} u}$ the Jacobi 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 the integral of the secant function. Using Cayley's notation,

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

He then derives "the definition of the transcendent",

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

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

The Gudermannian and its inverse were used to make trigonometric 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 ${\displaystyle \operatorname {gd} ^{-1}}$ column of the trigonometric table.[22]

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 hyperboloid of two sheets can be likewise mapped onto a n-dimensional hemisphere via stereographic projection. The hemisphere model of hyperbolic space uses such a map to represent hyperbolic space.

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

• The angle of parallelism function in hyperbolic geometry is the complement of the gudermannian, ${\displaystyle {\mathit {\Pi }}(\psi )={\tfrac {1}{2}}\pi -\operatorname {gd} \psi .}$
• On a Mercator projection a line of constant latitude is parallel to the equator (on the projection) and is displaced by an amount proportional to the inverse Gudermannian of the latitude.
• The Gudermannian (with a complex argument) may be used to define the transverse Mercator projection.[23]

• The Gudermannian appears in a non-periodic solution of the inverted pendulum.[24]
• The Gudermannian appears in a moving mirror solution of the dynamical Casimir effect.[25]
• If an infinite number of infinitely long, equidistant, parallel, coplanar, straight wires are kept at equal potentials with alternating signs, the potential-flux distribution in a cross-sectional plane perpendicular to the wires is the complex Gudermannian.[26]
• The Gudermannian function is a sigmoid function, and as such is sometimes used as an activation function in machine learning.
• The (scaled and shifted) Gudermannian is the cumulative distribution function of the hyperbolic secant distribution.
• A function based on the Gudermannian provides a good model for the shape of spiral galaxy arms.[27]

• Tractrix
• Catenary § Catenary of equal strength

• Penn, Michael (2020) "the Gudermannian function!" on YouTube.