Английская Википедия:Gudermannian function

The Gudermannian function relates the area of a circular sector to the area of a hyperbolic sector, via a common stereographic projection. If twice the area of the blue hyperbolic sector is Шаблон:Math, then twice the area of the red circular sector is Шаблон:Math. Twice the area of the purple triangle is the stereographic projection Шаблон:Math The blue point has coordinates Шаблон:Math. The red point has coordinates Шаблон:Math The purple point has coordinates Шаблон:Math

Файл:Gudermannian graph.png

Graph of the Gudermannian function.

Файл:Inverse Gudermannian graph.png

Graph of the inverse Gudermannian function.

In mathematics, the Gudermannian function relates a hyperbolic angle measure <math display=inline>\psi</math> to a circular angle measure <math display=inline>\phi</math> called the gudermannian of <math display=inline>\psi</math> and denoted <math display=inline>\operatorname{gd}\psi</math>.^[1] The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert, and later named for Christoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830.^[2] The gudermannian is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude <math display=inline>\operatorname{am}(\psi, m)</math> when parameter <math display=inline>m=1.</math>

The real Gudermannian function is typically defined for <math display=inline>-\infty < \psi < \infty</math> to be the integral of the hyperbolic secant^[3]

<math>

\phi = \operatorname{gd} \psi \equiv \int_0^\psi \operatorname{sech} t \,\mathrm{d}t = \operatorname{arctan} (\sinh \psi).</math>

The real inverse Gudermannian function can be defined for <math display=inline>-\tfrac12\pi < \phi < \tfrac12\pi</math> as the integral of the (circular) secant

<math>

\psi = \operatorname{gd}^{-1} \phi = \int_0^\phi \operatorname{sec} t \,\mathrm{d}t = \operatorname{arsinh} (\tan \phi). </math>

The hyperbolic angle measure <math>\psi = \operatorname{gd}^{-1} \phi</math> is called the anti-gudermannian of <math>\phi</math> or sometimes the lambertian of <math>\phi</math>, denoted <math>\psi = \operatorname{lam} \phi.</math>^[4] In the context of geodesy and navigation for latitude <math display=inline>\phi</math>, <math>k \operatorname{gd}^{-1} \phi</math> (scaled by arbitrary constant <math display=inline>k</math>) was historically called the meridional part of <math>\phi</math> (French: latitude croissante). It is the vertical coordinate of the Mercator projection.

The two angle measures <math display=inline>\phi</math> and <math display=inline>\psi</math> are related by a common stereographic projection

<math>s = \tan \tfrac12 \phi = \tanh \tfrac12 \psi,</math>

and this identity can serve as an alternative definition for <math display=inline>\operatorname{gd}</math> and <math display=inline>\operatorname{gd}^{-1}</math> valid throughout the complex plane:

<math>\begin{aligned}

\operatorname{gd} \psi &= {2\arctan}\bigl(\tanh\tfrac12 \psi \,\bigr), \\[5mu] \operatorname{gd}^{-1} \phi &= {2\operatorname{artanh}}\bigl(\tan\tfrac12 \phi \,\bigr). \end{aligned}</math>

Circular–hyperbolic identities

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

<math>\begin{align}

\operatorname{gd} \psi &\equiv \int_0^\psi \frac{1}{\operatorname{cosh} t}\mathrm{d}t = \int_0^{\tanh\frac12\psi} \frac{1-u^2}{1 + u^2}\frac{2\,\mathrm{d}u}{1 - u^2} \qquad \bigl(u = \tanh\tfrac12 t \bigr) \\[8mu] &= 2\int_0^{\tanh\frac12\psi} \frac{1}{1 + u^2} \mathrm{d}u = {2\arctan}\bigl(\tanh\tfrac12\psi\,\bigr), \\[5mu] \tan\tfrac12{\operatorname{gd} \psi} &= \tanh\tfrac12\psi. \end{align}</math>

Letting <math display=inline>\phi = \operatorname{gd} \psi</math> and <math display=inline>s = \tan \tfrac12 \phi = \tanh \tfrac12 \psi</math> we can derive a number of identities between hyperbolic functions of <math display=inline>\psi</math> and circular functions of <math display=inline>\phi.</math>^[6]

Identities related to the Gudermannian function represented graphically.

<math>\begin{align}

s &= \tan \tfrac12 \phi = \tanh \tfrac12 \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{align}</math>

These are commonly used as expressions for <math>\operatorname{gd}</math> and <math>\operatorname{gd}^{-1}</math> for real values of <math>\psi</math> and <math>\phi</math> with <math>|\phi| < \tfrac12\pi.</math> For example, the numerically well-behaved formulas

<math>\begin{align}

\operatorname{gd} \psi &= \operatorname{arctan} (\sinh \psi), \\[6mu] \operatorname{gd}^{-1} \phi &= \operatorname{arsinh} (\tan \phi). \end{align}</math>

(Note, for <math>|\phi| > \tfrac12\pi</math> and for complex arguments, care must be taken choosing branches of the inverse functions.)^[7]

We can also express <math display=inline>\psi</math> and <math display=inline>\phi</math> in terms of <math display=inline>s\colon</math>

<math>\begin{align}

2\arctan s &= \phi = \operatorname{gd} \psi, \\[6mu] 2\operatorname{artanh} s &= \operatorname{gd}^{-1} \phi = \psi. \\[6mu] \end{align}</math>

If we expand <math display=inline>\tan\tfrac12</math> and <math display=inline>\tanh\tfrac12</math> in terms of the exponential, then we can see that <math display=inline>s,</math> <math>\exp \phi i,</math> and <math>\exp \psi</math> are all Möbius transformations of each-other (specifically, rotations of the Riemann sphere):

<math>\begin{align}

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{align}</math>

For real values of <math display=inline>\psi</math> and <math display=inline>\phi</math> with <math>|\phi| < \tfrac12\pi</math>, these Möbius transformations can be written in terms of trigonometric functions in several ways,

<math>\begin{align}

\exp \psi &= \sec \phi + \tan \phi = \tan\tfrac12 \bigl(\tfrac12\pi + \phi \bigr) \\[6mu] &= \frac{1 + \tan\tfrac12 \phi}{1 - \tan\tfrac12 \phi} = \sqrt{\frac{1+\sin \phi}{1-\sin \phi}}, \\[12mu]

\exp \phi i &= \operatorname{sech} \psi + i \tanh \psi = \tanh\tfrac12 \bigl({-\tfrac12}\pi i + \psi \bigr) \\[6mu] &= \frac{1 + i \tanh\tfrac12 \psi}{1 - i \tanh\tfrac12 \psi} = \sqrt{\frac{1 + i \sinh \psi}{1 - i \sinh \psi}}. \end{align}</math>

These give further expressions for <math>\operatorname{gd}</math> and <math>\operatorname{gd}^{-1}</math> for real arguments with <math>|\phi| < \tfrac12\pi.</math> For example,^[8]

<math>\begin{align}

\operatorname{gd} \psi &= 2 \arctan e^\psi - \tfrac12\pi, \\[6mu] \operatorname{gd}^{-1} \phi &= \log (\sec \phi + \tan \phi). \end{align}</math>

Complex values

Файл:Gudermannian conformal map.png

The Gudermannian function Шаблон:Math is a conformal map from an infinite strip to an infinite strip. It can be broken into two parts: a map Шаблон:Math from one infinite strip to the complex unit disk and a map Шаблон:Math from the disk to the other infinite strip.

As a functions of a complex variable, <math display=inline>z \mapsto w = \operatorname{gd} z</math> conformally maps the infinite strip <math display=inline>\left|\operatorname{Im}z\right| \leq \tfrac12\pi</math> to the infinite strip <math display=inline>\left|\operatorname{Re}w\right| \leq \tfrac12\pi,</math> while <math display=inline>w \mapsto z = \operatorname{gd}^{-1} w</math> conformally maps the infinite strip <math display=inline>\left|\operatorname{Re}w\right| \leq \tfrac12\pi</math> to the infinite strip <math display=inline> \left|\operatorname{Im}z\right| \leq \tfrac12\pi.</math>

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

<math>\begin{aligned}

\operatorname{gd} z &= {2\arctan}\bigl(\tanh\tfrac12 z \,\bigr), \\[5mu] \operatorname{gd}^{-1} w &= {2\operatorname{artanh}}\bigl(\tan\tfrac12 w \,\bigr). \end{aligned}</math>

For the <math display=inline>\operatorname{gd}</math> and <math display=inline>\operatorname{gd}^{-1}</math> functions to remain invertible with these extended domains, we might consider each to be a multivalued function (perhaps <math display=inline>\operatorname{Gd}</math> and <math display=inline>\operatorname{Gd}^{-1}</math>, with <math display=inline>\operatorname{gd}</math> and <math display=inline>\operatorname{gd}^{-1}</math> the principal branch) or consider their domains and codomains as Riemann surfaces.

If <math display=inline>u + iv = \operatorname{gd}(x + iy),</math> then the real and imaginary components <math display=inline>u</math> and <math display=inline>v</math> can be found by:^[10]

<math>

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

(In practical implementation, make sure to use the 2-argument arctangent, {{{1}}}

Likewise, if <math display=inline>x + iy = \operatorname{gd}^{-1}(u + iv),</math> then components <math display=inline>x</math> and <math display=inline>y</math> can be found by:^[11]

<math>

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

Multiplying these together reveals the additional identity^[8]

<math>

\tanh x\, \tan y = \tan u\, \tanh v. </math>

Symmetries

The two functions can be thought of as rotations or reflections of each-other, with a similar relationship as <math display=inline>\sinh iz = i \sin z</math> between sine and hyperbolic sine:^[12]

<math>\begin{aligned}

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

The functions are both odd and they commute with complex conjugation. That is, a reflection across the real or imaginary axis in the domain results in the same reflection in the codomain:

<math>\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}</math>

The functions are periodic, with periods <math display=inline>2\pi i</math> and <math display=inline>2\pi</math>:

<math>\begin{aligned}

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

A translation in the domain of <math display=inline>\operatorname{gd}</math> by <math display=inline>\pm\pi i</math> results in a half-turn rotation and translation in the codomain by one of <math display=inline>\pm\pi,</math> and vice versa for <math display=inline>\operatorname{gd}^{-1}\colon</math>^[13]

<math>\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}</math>

A reflection in the domain of <math display=inline>\operatorname{gd}</math> across either of the lines <math display=inline>x \pm \tfrac12\pi i</math> results in a reflection in the codomain across one of the lines <math display=inline>\pm \tfrac12\pi + yi,</math> and vice versa for <math display=inline>\operatorname{gd}^{-1}\colon</math>

<math>\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}</math>

This is related to the identity

<math>

\tanh\tfrac12 ({\pi i} \pm z) = \tan\tfrac12 ({\pi} \mp \operatorname{gd} z). </math>

Specific values

A few specific values (where <math display=inline>\infty</math> indicates the limit at one end of the infinite strip):^[14]

<math>\begin{align}

\operatorname{gd}(0) &= 0, &\quad

 {\operatorname{gd}}\bigl({\pm {\log}\bigl(2 + \sqrt3\bigr)}\bigr) &= \pm \tfrac13\pi, \\[5mu]

\operatorname{gd}(\pi i) &= \pi, &\quad

 {\operatorname{gd}}\bigl({\pm \tfrac13}\pi i\bigr) &= \pm {\log}\bigl(2 + \sqrt3\bigr)i, \\[5mu]

\operatorname{gd}({\pm \infty}) &= \pm\tfrac12\pi, &\quad

 {\operatorname{gd}}\bigl({\pm {\log}\bigl(1 + \sqrt2\bigr)}\bigr) &= \pm \tfrac14\pi, \\[5mu]

{\operatorname{gd}}\bigl({\pm \tfrac12}\pi i\bigr) &= \pm \infty i, &\quad

 {\operatorname{gd}}\bigl({\pm \tfrac14}\pi i\bigr) &= \pm {\log}\bigl(1 + \sqrt2\bigr)i, \\[5mu]

&& {\operatorname{gd}}\bigl({\log}\bigl(1 + \sqrt2\bigr) \pm \tfrac12\pi i\bigr) &= \tfrac12\pi \pm {\log}\bigl(1 + \sqrt2\bigr)i, \\[5mu] && {\operatorname{gd}}\bigl({-\log}\bigl(1 + \sqrt2\bigr) \pm \tfrac12\pi i\bigr) &= -\tfrac12\pi \pm {\log}\bigl(1 + \sqrt2\bigr)i.

\end{align}</math>

Derivatives

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:

<math>\begin{align}

\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{align}</math>

Argument-addition identities

By combining hyperbolic and circular argument-addition identities,

<math>\begin{align}

\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{align}</math>

with the circular–hyperbolic identity,

<math>

\tan \tfrac12 (\operatorname{gd} z) = \tanh \tfrac12 z, </math>

we have the Gudermannian argument-addition identities:

<math>\begin{align}

\operatorname{gd}(z + w) &= 2 \arctan \frac

 {\tan \tfrac12(\operatorname{gd} z) + \tan\tfrac12(\operatorname{gd} w)}
 {1 + \tan\tfrac12(\operatorname{gd} z) \, \tan\tfrac12(\operatorname{gd} w)}, \\[12mu]

\operatorname{gd}^{-1}(z + w) &= 2 \operatorname{artanh} \frac

 {\tanh\tfrac12(\operatorname{gd}^{-1} z) + \tanh\tfrac12(\operatorname{gd}^{-1} w)}
 {1 - \tanh\tfrac12(\operatorname{gd}^{-1} z) \, \tanh\tfrac12(\operatorname{gd}^{-1} w)}.

\end{align}</math>

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,

<math>\begin{align}

\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{align}</math>

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

In the specific case <math display=inline>z = w,</math> double-argument identities are

<math>\begin{align}

\operatorname{gd}(2z) &= 2 \arctan (\sin(\operatorname{gd} z)), \\[5mu] \operatorname{gd}^{-1}(2z) &= 2 \operatorname{artanh}(\sinh(\operatorname{gd}^{-1}z)). \end{align}</math>

Taylor series

The Taylor series near zero, valid for complex values <math display=inline>z</math> with <math display=inline>|z| < \tfrac12\pi,</math> are^[17]

<math>\begin{align}

\operatorname{gd} z &= \sum_{k=0}^\infty \frac{E_k}{(k+1)!}z^{k+1} = z - \frac16z^3 + \frac1{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 + \frac16z^3 + \frac1{24}z^5 + \frac{61}{5040}z^7 + \frac{277}{72576}z^9 + \dots, \end{align}</math>

where the numbers <math display=inline>E_{k}</math> are the Euler secant numbers, 1, 0, -1, 0, 5, 0, -61, 0, 1385 ... (sequences Шаблон:OEIS link, Шаблон:OEIS link, and Шаблон:OEIS link 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 <math display=inline>E_{k}</math> and <math display=inline>|E_{k}|</math> are same as the numerators of the [[Hyperbolic functions#Taylor series expressions|Taylor series for Шаблон:Math]] and [[Trigonometric functions#Power series expansion|Шаблон:Math]], 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 Шаблон:OEIS link and Шаблон:OEIS link in the OEIS).

History

Шаблон:Broader 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 <math display=inline>\psi.</math> In terms of latitude <math display=inline>\phi</math> on the sphere (expressed in radians) the isometric latitude can be written

<math>\psi = \operatorname{gd}^{-1} \phi = \int_0^\phi \sec t \,\mathrm{d}t.</math>

The inverse from the isometric latitude to spherical latitude is <math display=inline>\phi = \operatorname{gd} \psi.</math> (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 <math display=inline>\sinh</math> and <math display=inline>\cosh</math> to a wide audience (although represented by the symbols <math display=inline>\mathfrak{Sin}</math> and <math display=inline>\mathfrak{Cos}</math>).

The notation <math display=inline>\operatorname{gd}</math> was introduced by Cayley who starts by calling <math display=inline>\phi = \operatorname{gd} u</math> the Jacobi elliptic amplitude <math display=inline>\operatorname{am} u</math> in the degenerate case where the elliptic modulus is <math display=inline>m = 1,</math> so that <math display=inline>\sqrt{1 + m\sin\!^2\,\phi}</math> reduces to <math display=inline>\cos \phi.</math>^[21] This is the inverse of the integral of the secant function. Using Cayley's notation,

<math>

u = \int_0 \frac{d\phi}{\cos \phi} = {\log\, \tan}\bigl(\tfrac14\pi + \tfrac12 \phi\bigr). </math>

He then derives "the definition of the transcendent",

<math>

\operatorname{gd} u = {\frac1i \log\, \tan} \bigl(\tfrac14\pi + \tfrac12 ui\bigr), </math>

observing that "although exhibited in an imaginary form, [it] is a real function of {{{1}}}

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 <math display=inline>\psi</math>, hyperbolic functions could be found by first looking up <math display=inline>\phi = \operatorname{gd} \psi</math> in a Gudermannian table and then looking up the appropriate circular function of <math display=inline>\phi</math>, or by directly locating <math display=inline>\psi</math> in an auxiliary <math>\operatorname{gd}^{-1}</math> column of the trigonometric table.^[22]

Generalization

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.

Applications

Файл:Distance in the half-plane model 3.png

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, <math>\mathit{\Pi}(\psi) = \tfrac12\pi - \operatorname{gd} \psi.</math>
On a Mercator 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 a complex argument) may be used to define the transverse Mercator projection.^[23]
The Gudermannian function appears in a non-periodic solution of the inverted pendulum.^[24]
The Gudermannian function 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 function.^[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 function 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]

Notes

Шаблон:Reflist

References

Шаблон:Sfn whitelist Шаблон:Refbegin

Шаблон:Refend

External links

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

Шаблон:Bots

↑ The symbols <math display=inline>\psi</math> and <math display=inline>\phi</math> were chosen for this article because they are commonly used in geodesy for the isometric latitude (vertical coordinate of the Mercator projection) and geodetic latitude, respectively, and geodesy/cartography was the original context for the study of the Gudermannian and inverse Gudermannian functions.
↑ Gudermann published several papers about the trigonometric and hyperbolic functions in Crelle's Journal in 1830–1831. These were collected in a book, Шаблон:Harvp.
↑ Шаблон:Harvp §4.23(viii) "Gudermannian Function"; Шаблон:Harvp
↑ Шаблон:Harvp; Шаблон:Harvp
↑ Шаблон:Harvp
↑ Шаблон:Harvp pp. 23–27
↑ Шаблон:Harvp draws complex-valued plots of several of these, demonstrating that naïve implementations that choose the principal branch of inverse trigonometric functions yield incorrect results.
↑ ^8,0 ^8,1 Шаблон:Mathworld
↑ Шаблон:Harvp
↑ Шаблон:Harvp p. 181; Шаблон:Harvp p. 269
↑ Шаблон:Harvp p. 269 – note the typo.
↑ Шаблон:Harvp §4.2.8(163) pp. 144–145
↑ Шаблон:Harvp p. 182
↑ Шаблон:Harvp
↑ Шаблон:Harvp p. 21
↑ Шаблон:Harvp pp. 180–183
↑ Шаблон:Harvp §4.2.7(162) pp. 143–144
↑ Шаблон:Cite book
↑ Шаблон:Harvp
↑ Шаблон:Harvp
↑ Шаблон:Harvp
↑ For example Hoüel labels the hyperbolic functions across the top in Table XIV of: Шаблон:Cite book
↑ Шаблон:Harvp p. 74
↑ Шаблон:Harvp
↑ Шаблон:Harvp
↑ Шаблон:Harvp
↑ Шаблон:Harvp

[1] The symbols <math display=inline>\psi</math> and <math display=inline>\phi</math> were chosen for this article because they are commonly used in geodesy for the isometric latitude (vertical coordinate of the Mercator projection) and geodetic latitude, respectively, and geodesy/cartography was the original context for the study of the Gudermannian and inverse Gudermannian functions.

[2] Gudermann published several papers about the trigonometric and hyperbolic functions in Crelle's Journal in 1830–1831. These were collected in a book, Шаблон:Harvp.

[3] Шаблон:Harvp §4.23(viii) "Gudermannian Function"; Шаблон:Harvp

[4] Шаблон:Harvp; Шаблон:Harvp

[5] Шаблон:Harvp

[6] Шаблон:Harvp pp. 23–27

[7] Шаблон:Harvp draws complex-valued plots of several of these, demonstrating that naïve implementations that choose the principal branch of inverse trigonometric functions yield incorrect results.

[weinstein-8] 8,0 ^8,1 Шаблон:Mathworld

[9] Шаблон:Harvp

[10] Шаблон:Harvp p. 181; Шаблон:Harvp p. 269

[11] Шаблон:Harvp p. 269 – note the typo.

[12] Шаблон:Harvp §4.2.8(163) pp. 144–145

[13] Шаблон:Harvp p. 182

[14] Шаблон:Harvp

[15] Шаблон:Harvp p. 21

[16] Шаблон:Harvp pp. 180–183

[17] Шаблон:Harvp §4.2.7(162) pp. 143–144

[18] Шаблон:Cite book

[19] Шаблон:Harvp

[20] Шаблон:Harvp

[21] Шаблон:Harvp

[22] For example Hoüel labels the hyperbolic functions across the top in Table XIV of: Шаблон:Cite book

[23] Шаблон:Harvp p. 74

[24] Шаблон:Harvp

[25] Шаблон:Harvp

[26] Шаблон:Harvp

[27] Шаблон:Harvp

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

Партнерские ресурсы
Криптовалюты	Обмен криптовалют - www.bestchange.ru Криптовалютная биржа CoinEx Криптовалютная биржа Binance HIVE OS - операционная система для майнинга e4pool - Мультивалютный пул для майнинга.
Магазины	AliExpress — глобальная виртуальная (в Интернете) торговая площадка, предоставляющая возможность покупать товары производителей из КНР; computeruniverse.net - Интернет-магазин компьютеров(Промо код 5 Евро на первую покупку:FWWC3ZKQ);
Хостинг	DigitalOcean - американский провайдер облачных инфраструктур, с главным офисом в Нью-Йорке и с центрами обработки данных по всему миру;
Разное	Викиум - Онлайн-тренажер для мозга Like Центр - Центр поддержки и развития предпринимательства. Gamersbay - лучший магазин по бустингу для World of Warcraft. Ноотропы OmniMind N°1 - Усиливает мозговую активность. Повышает мотивацию. Улучшает память. Санкт-Петербургская школа телевидения - это федеральная сеть образовательных центров, которая имеет филиалы в 37 городах России. Lingualeo.com — интерактивный онлайн-сервис для изучения и практики английского языка в увлекательной игровой форме. Junyschool (Джунискул) – международная школа программирования и дизайна для детей и подростков от 5 до 17 лет, где ученики осваивают компьютерную грамотность, развивают алгоритмическое и креативное мышление, изучают основы программирования и компьютерной графики, создают собственные проекты: игры, сайты, программы, приложения, анимации, 3D-модели, монтируют видео. Умназия - Интерактивные онлайн-курсы и тренажеры для развития мышления детей 6-13 лет SkillBox - это один из лидеров российского рынка онлайн-образования. Среди партнеров Skillbox ведущий разработчик сервисного дизайна AIC, медиа-компания Yoola, первое и самое крупное русскоязычное аналитическое агентство Tagline, онлайн-школа дизайна и иллюстрации Bang! Bang! Education, оператор PR-рынка PACO, студия рисования Draw&Go, агентство performance-маркетинга Ingate, scrum-студия Sibirix, имидж-лаборатория Персона. «Нетология» — это университет по подготовке и дополнительному обучению специалистов в области интернет-маркетинга, управления проектами и продуктами, дизайна, Data Science и разработки. В рамках Нетологии студенты получают ценные теоретические знания от лучших экспертов Рунета, выполняют практические задания на отработку полученных навыков, общаются с экспертами и единомышленниками. Познакомиться со всеми продуктами подробнее можно на сайте https://netology.ru, линейка курсов и профессий постоянно обновляется. StudyBay Brazil – это онлайн биржа для португалоговорящих студентов и авторов! Студент получает уникальную работу любого уровня сложности и больше свободного времени, в то время как у автора появляется дополнительный заработок и бесценный опыт. Автор24 — самая большая в России площадка по написанию учебных работ: контрольные и курсовые работы, дипломы, рефераты, решение задач, отчеты по практике, а так же любой другой вид работы. Сервис сотрудничает с более 70 000 авторов. Более 1 000 000 работ уже выполнено. StudyBay – это онлайн биржа для англоязычных студентов и авторов! Студент получает уникальную работу любого уровня сложности и больше свободного времени, в то время как у автора появляется дополнительный заработок и бесценный опыт.

Английская Википедия:Gudermannian function

Содержание

Circular–hyperbolic identities

Complex values

Symmetries

Specific values

Derivatives

Argument-addition identities

Taylor series

History

Generalization

Applications

See also

Notes

References

External links

Навигация

Действия на странице

Действия на странице

Персональные инструменты

Навигация

Поиск

Инструменты