Gudermannian function explained

In mathematics, the Gudermannian function relates a hyperbolic angle measure $\backslash psi$ to a circular angle measure $\backslash phi$ called the gudermannian of $\backslash psi$ and denoted $\backslash operatorname\backslash psi$.^[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 $\backslash operatorname(\backslash psi,\; m)$ when parameter $m=1.$

The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant^[3]

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

The real inverse Gudermannian function can be defined for as the integral of the (circular) secant

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

The hyperbolic angle measure

\psi = \operatorname{gd} ^{-1} \phi is called the anti-gudermannian of \phi or sometimes the lambertian of \phi , denoted \psi = \operatorname{lam} \phi. In the context of geodesy and navigation for latitude $\backslash phi$, k \operatorname{gd} ^{-1} \phi (scaled by arbitrary constant $k$) was historically called the meridional part of \phi (French: latitude croissante). It is the vertical coordinate of the Mercator projection.

The two angle measures $\backslash phi$ and $\backslash psi$ are related by a common stereographic projection

s = \tan \tfrac 12 \phi = \tanh \tfrac 12 \psi,

and this identity can serve as an alternative definition for $\backslash operatorname$ and $\backslash operatorname^$ valid throughout the complex plane:

\begin{aligned} \operatorname{gd} \psi & = { 2 \arctan} l (\tanh\tfrac 12 \psi r ), \\[ 5 mu] \operatorname{gd} ^{-1} \phi & = { 2 \operatorname{artanh}} l (\tan\tfrac 12 \phi r ). \end{aligned}

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Circular–hyperbolic identities

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

\begin{align} \operatorname{gd} \psi &\equiv \int ^{\psi} _{0} (1)/(\operatorname{cosh) t} d t = \int ^{\tanh (12 \psi)/()} _{0} (1- u ^{2})/(1+ u ^{2}) (2 d u)/(1- u ^{2}) l (u = \tanh\tfrac 12 t r ) \\[ 8 mu] & = 2 \int ^{\tanh (12 \psi)/()} _{0} (1)/(1+ u ^{2}) d u = { 2 \arctan} l (\tanh\tfrac 12 \psi r ), \\[ 5 mu] \tan\tfrac 12 {\operatorname{gd} \psi} & = \tanh\tfrac 12 \psi. \end{align}

Letting $\backslash phi\; =\; \backslash operatorname\; \backslash psi$ and $s\; =\; \backslash tan\; \backslash tfrac12\; \backslash phi\; =\; \backslash tanh\; \backslash tfrac12\; \backslash psi$ we can derive a number of identities between hyperbolic functions of $\backslash psi$ and circular functions of $\backslash phi.$^[4]

\begin{align} s & = \tan \tfrac 12 \phi = \tanh \tfrac 12 \psi, \\[ 6 mu] (2 s)/(1+ s ^{2}) & = \sin \phi = \tanh \psi, & (1+ s ^{2})/(2 s) & = \csc \phi = \coth \psi, \\[ 10 mu] (1- s ^{2})/(1+ s ^{2}) & = \cos \phi = \operatorname{sech} \psi, & (1+ s ^{2})/(1- s ^{2}) & = \sec \phi = \cosh \psi, \\[ 10 mu] (2 s)/(1- s ^{2}) & = \tan \phi = \sinh \psi, & (1- s ^{2})/(2 s) & = \cot \phi = \operatorname{csch} \psi. \\[ 8 mu] \end{align}

These are commonly used as expressions for

\operatorname{gd} and \operatorname{gd} ^{-1} for real values of \psi and \phi with |\phi| < \tfrac 12 \pi. For example, the numerically well-behaved formulas\begin{align} \operatorname{gd} \psi & = \operatorname{arctan} (\sinh \psi), \\[ 6 mu] \operatorname{gd} ^{-1} \phi & = \operatorname{arsinh} (\tan \phi). \end{align}

(Note, for

|\phi| > \tfrac 12 \pi and for complex arguments, care must be taken choosing branches of the inverse functions.)^[5]

We can also express $\backslash psi$ and $\backslash phi$ in terms of $s\backslash colon$

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

If we expand $\backslash tan\backslash tfrac12$ and $\backslash tanh\backslash tfrac12$ in terms of the exponential, then we can see that $s,$

\exp \phi i, and \exp \psi are all Möbius transformations of each-other (specifically, rotations of the Riemann sphere):\begin{align} s & = i (1- e ^{\phi})/(1+ e ^{\phi}) = (e ^{\psi} -1)/(e ^{\psi} +1) , \\[ 10 mu] i (s - i)/(s + i) & = \exp \phi i = (e ^{\psi} - i)/(e ^{\psi} + i) , \\[ 10 mu] (1+ s)/(1- s) & = i (i + e ^{\phi})/(i - e ^{\phi}) = \exp \psi. \end{align}

For real values of $\backslash psi$ and $\backslash phi$ with

|\phi| < \tfrac 12 \pi , these Möbius transformations can be written in terms of trigonometric functions in several ways,\begin{align} \exp \psi & = \sec \phi + \tan \phi = \tan\tfrac 12 l (\tfrac 12 \pi + \phi r ) \\[ 6 mu] & = (1+ \tan\tfrac 12 \phi)/(1- \tan\tfrac 12 \phi) = \sqrt{ (1+ \sin \phi)/(1- \sin \phi) }, \\[12mu]

\exp \phi i&= \operatorname \psi + i \tanh \psi= \tanh\tfrac12 \bigl(\pi i + \psi \bigr) \\[6mu]&= \frac= \sqrt.\end

These give further expressions for

\operatorname{gd} and \operatorname{gd} ^{-1} for real arguments with |\phi| < \tfrac 12 \pi. For example,\begin{align} \operatorname{gd} \psi & = 2 \arctan e ^{\psi} - \tfrac 12 \pi, \\[ 6 mu] \operatorname{gd} ^{-1} \phi & = log (\sec \phi + \tan \phi). \end{align}

Complex values

As a function of a complex variable, $z\; \backslash mapsto\; w\; =\; \backslash operatorname\; z$ conformally maps the infinite strip $\backslash left|\backslash operatornamez\backslash right|\; \backslash leq\; \backslash tfrac12\backslash pi$ to the infinite strip $\backslash left|\backslash operatornamew\backslash right|\; \backslash leq\; \backslash tfrac12\backslash pi,$ while $w\; \backslash mapsto\; z\; =\; \backslash operatorname^\; w$ conformally maps the infinite strip $\backslash left|\backslash operatornamew\backslash right|\; \backslash leq\; \backslash tfrac12\backslash pi$ to the infinite strip $\backslash left|\backslash operatornamez\backslash right|\; \backslash leq\; \backslash tfrac12\backslash pi.$

Analytically continued by reflections to the whole complex plane, $z\; \backslash mapsto\; w\; =\; \backslash operatorname\; z$ is a periodic function of period $2\backslash pi\; i$ which sends any infinite strip of "height" $2\backslash pi\; i$ onto the strip Likewise, extended to the whole complex plane, $w\; \backslash mapsto\; z\; =\; \backslash operatorname^\; w$ is a periodic function of period $2\backslash pi$ which sends any infinite strip of "width" $2\backslash pi$ onto the strip For all points in the complex plane, these functions can be correctly written as:

\begin{aligned} \operatorname{gd} z & = { 2 \arctan} l (\tanh\tfrac 12 z r ), \\[ 5 mu] \operatorname{gd} ^{-1} w & = { 2 \operatorname{artanh}} l (\tan\tfrac 12 w r ). \end{aligned}

For the $\backslash operatorname$ and $\backslash operatorname^$ functions to remain invertible with these extended domains, we might consider each to be a multivalued function (perhaps $\backslash operatorname$ and $\backslash operatorname^$, with $\backslash operatorname$ and $\backslash operatorname^$ the principal branch) or consider their domains and codomains as Riemann surfaces.

If $u\; +\; iv\; =\; \backslash operatorname(x\; +\; iy),$ then the real and imaginary components $u$ and $v$ can be found by:^[6]

\tan u = (\sinh x)/(\cos y) , \tanh v = (\sin y)/(\cosh x) .

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

Likewise, if $x\; +\; iy\; =\; \backslash operatorname^(u\; +\; iv),$ then components $x$ and $y$ can be found by:^[7]

\tanh x = (\sin u)/(\cosh v) , \tan y = (\sinh v)/(\cos u) .

Multiplying these together reveals the additional identity

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

Symmetries

The two functions can be thought of as rotations or reflections of each-other, with a similar relationship as $\backslash sinh\; iz\; =\; i\; \backslash sin\; z$ between sine and hyperbolic sine:^[8]

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

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:

\begin{aligned} \operatorname{gd} ( - z) & = - \operatorname{gd} z, & \operatorname{gd} \bar z & = \overline{\operatorname{gd} z}, & \operatorname{gd} ( - \bar z) & = - \overline{\operatorname{gd} z}, \\[ 5 mu] \operatorname{gd} ^{-1} ( - z) & = - \operatorname{gd} ^{-1} z, & \operatorname{gd} ^{-1} \bar z & = \overline{\operatorname{gd} ^{-1} z}, & \operatorname{gd} ^{-1} ( - \bar z) & = - \overline{\operatorname{gd} ^{-1} z}. \end{aligned}

The functions are periodic, with periods $2\backslash pi\; i$ and $2\backslash pi$:

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

A translation in the domain of $\backslash operatorname$ by $\backslash pm\backslash pi\; i$ results in a half-turn rotation and translation in the codomain by one of $\backslash pm\backslash pi,$ and vice versa for $\backslash operatorname^\backslash colon$^[9]

\begin{aligned} \operatorname{gd} ({\pm \pi i} + z) & = \begin{cases} \pi - \operatorname{gd} z & if \operatorname{Re} z \geq 0 , \\[ 5 mu] - \pi - \operatorname{gd} z & if \operatorname{Re} z < 0 , \end{cases} \\[ 15 mu] \operatorname{gd} ^{-1} ({\pm \pi} + z) & = \begin{cases} \pi i - \operatorname{gd} ^{-1} z & if \operatorname{Im} z \geq 0 , \\[ 3 mu] - \pi i - \operatorname{gd} ^{-1} z & if \operatorname{Im} z < 0 . \end{cases} \end{aligned}

A reflection in the domain of $\backslash operatorname$ across either of the lines $x\; \backslash pm\; \backslash tfrac12\backslash pi\; i$ results in a reflection in the codomain across one of the lines $\backslash pm\; \backslash tfrac12\backslash pi\; +\; yi,$ and vice versa for $\backslash operatorname^\backslash colon$

\begin{aligned} \operatorname{gd} ({\pm \pi i} + \bar z) & = \begin{cases} \pi - \overline{\operatorname{gd} z} & if \operatorname{Re} z \geq 0 , \\[ 5 mu] - \pi - \overline{\operatorname{gd} z} & if \operatorname{Re} z < 0 , \end{cases} \\[ 15 mu] \operatorname{gd} ^{-1} ({\pm \pi} - \bar z) & = \begin{cases} \pi i + \overline{\operatorname{gd} ^{-1} z} & if \operatorname{Im} z \geq 0 , \\[ 3 mu] - \pi i + \overline{\operatorname{gd} ^{-1} z} & if \operatorname{Im} z < 0 . \end{cases} \end{aligned}

This is related to the identity

\tanh\tfrac 12 ({\pi i} \pm z) = \tan\tfrac 12 ({\pi} \mp \operatorname{gd} z).

Specific values

A few specific values (where $\backslash infty$ indicates the limit at one end of the infinite strip):

\begin{align} \operatorname{gd}( 0 ) & = 0 , & {\operatorname{gd}} l ({\pm { log } l ( 2 + \sqrt 3 r )} r ) & = \pm \tfrac 13 \pi, \\[ 5 mu] \operatorname{gd}(\pi i) & = \pi, & {\operatorname{gd}} l ({\pm \tfrac 13 }\pi i r ) & = \pm { log } l ( 2 + \sqrt 3 r )i, \\[ 5 mu] \operatorname{gd}({\pm inf ty}) & = \pm\tfrac 12 \pi, & {\operatorname{gd}} l ({\pm { log } l ( 1 + \sqrt 2 r )} r ) & = \pm \tfrac 14 \pi, \\[ 5 mu] {\operatorname{gd}} l ({\pm \tfrac 12 }\pi i r ) & = \pm inf ty i, & {\operatorname{gd}} l ({\pm \tfrac 14 }\pi i r ) & = \pm { log } l ( 1 + \sqrt 2 r )i, \\[ 5 mu] && {\operatorname{gd}} l ({ log } l ( 1 + \sqrt 2 r ) \pm \tfrac 12 \pi i r ) & = \tfrac 12 \pi \pm { log } l ( 1 + \sqrt 2 r )i, \\[ 5 mu] && {\operatorname{gd}} l ({ -log } l ( 1 + \sqrt 2 r ) \pm \tfrac 12 \pi i r ) & = - \tfrac 12 \pi \pm { log } l ( 1 + \sqrt 2 r )i. \end{align}

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:

\begin{align} (d)/(d z) \operatorname{gd} z & = \operatorname{sech} z , \\[ 10 mu] (d)/(d z) \operatorname{gd} ^{-1} z & = \sec z . \end{align}

Argument-addition identities

By combining hyperbolic and circular argument-addition identities,

\begin{align} \tanh(z + w) & = (\tanh z + \tanh w)/(1+ \tanh z \tanh w) , \\[ 10 mu] \tan(z + w) & = (\tan z + \tan w)/(1- \tan z \tan w) , \end{align}

with the circular–hyperbolic identity,

\tan \tfrac 12 (\operatorname{gd} z) = \tanh \tfrac 12 z,

we have the Gudermannian argument-addition identities:

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

Further argument-addition identities can be written in terms of other circular functions,^[10] but they require greater care in choosing branches in inverse functions. Notably,

\begin{align} \operatorname{gd}(z + w) & = u + v, where \tan u = (\sinh z)/(\cosh w) , \tan v = (\sinh w)/(\cosh z) , \\[ 10 mu] \operatorname{gd} ^{-1} (z + w) & = u + v, where \tanh u = (\sin z)/(\cos w) , \tanh v = (\sin w)/(\cos z) , \end{align}

which can be used to derive the per-component computation for the complex Gudermannian and inverse Gudermannian.^[11]

In the specific case $z\; =\; w,$ double-argument identities are

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

Taylor series

The Taylor series near zero, valid for complex values $z$ with are^[12]

\begin{align} \operatorname{gd} z & = \sum ^{inf ty} _{k =0} (E _{k})/((k +1 )!) z ^{k+1} = z - (16 z)/(^{3}) + (1 { 24 }z)/(^{5}) - (61)/(5040) z ^{7} + (277)/(72576) z ^{9} - ... , \\[ 10 mu] \operatorname{gd} ^{-1} z & = \sum ^{inf ty} _{k =0} (|E _{k|})/((k +1 )!) z ^{k+1} = z + (16 z)/(^{3}) + (1 { 24 }z)/(^{5}) + (61)/(5040) z ^{7} + (277)/(72576) z ^{9} + ... , \end{align}

where the numbers $E\_$ are the Euler secant numbers, 1, 0, -1, 0, 5, 0, -61, 0, 1385 ... (sequences,, and in the OEIS). These series were first computed by James Gregory in 1671.^[13]

Because the Gudermannian and inverse Gudermannian functions are the integrals of the hyperbolic secant and secant functions, the numerators $E\_$ and $|E\_|$ are same as the numerators of the Taylor series for and, 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 and in the OEIS).

History

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 $\backslash psi.$ In terms of latitude $\backslash phi$ on the sphere (expressed in radians) the isometric latitude can be written

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

The inverse from the isometric latitude to spherical latitude is $\backslash phi\; =\; \backslash operatorname\; \backslash psi.$ (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.Gudermann had published articles in Crelle's Journal that were later collected in a bookwhich expounded $\backslash sinh$ and $\backslash cosh$ to a wide audience (although represented by the symbols $\backslash mathfrak$ and $\backslash mathfrak$).

The notation $\backslash operatorname$ was introduced by Cayley who starts by calling $\backslash phi\; =\; \backslash operatorname\; u$ the Jacobi elliptic amplitude $\backslash operatorname\; u$ in the degenerate case where the elliptic modulus is $m\; =\; 1,$ so that $\backslash sqrt$ reduces to $\backslash cos\; \backslash phi.$ This is the inverse of the integral of the secant function. Using Cayley's notation,

u = \int _{0} (d\phi)/(\cos \phi) = { log \tan} l (\tfrac 14 \pi + \tfrac 12 \phi r ).

He then derives "the definition of the transcendent",

\operatorname{gd} u = { (1 i)/(log) \tan} l (\tfrac 14 \pi + \tfrac 12 ui r ),

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

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 $\backslash psi$, hyperbolic functions could be found by first looking up $\backslash phi\; =\; \backslash operatorname\; \backslash psi$ in a Gudermannian table and then looking up the appropriate circular function of $\backslash phi$, or by directly locating $\backslash psi$ in an auxiliary

\operatorname{gd} ^{-1} column of the trigonometric table.^[14]

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

• The angle of parallelism function in hyperbolic geometry is the complement of the gudermannian,

\Pi (\psi) = \tfrac 12 \pi - \operatorname{gd} \psi.

• 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.^[15]
• The Gudermannian function appears in a non-periodic solution of the inverted pendulum.
• The Gudermannian function appears in a moving mirror solution of the dynamical Casimir effect.
• 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.
• 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.

See also

• Tractrix

References

• Barnett . Janet Heine . Janet Barnett . 2004 . Enter, Stage Center: The Early Drama of the Hyperbolic Functions . Mathematics Magazine . 77 . 1 . 15–30 . 10.1080/0025570X.2004.11953223 .
• Book: Becker . George Ferdinand . George Ferdinand Becker . Van Orstrand . Charles Edwin . 1909 . Hyperbolic Functions . Smithsonian Mathematical Tables . Smithsonian Institution .
• Becker . George Ferdinand . 1912 . The gudermannian complement and imaginary geometry . The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science . 24 . 142 . 600–608 . 10.1080/14786441008637363 .
• Book: Beyer . William H. . 1987 . CRC Handbook of Mathematical Sciences . 6th . 268–286 . CRC Press . limited .
• Cayley . Arthur . Arthur Cayley . 1862 . On the transcendent $\backslash operatorname\; u\; =\; \backslash tfrac\backslash log\; \backslash tan\; \backslash bigl(\backslash tfrac14\backslash pi\; +\; \backslash tfrac12\; ui\backslash bigr)$ . Philosophical Magazine . 4th Series . 24 . 158 . 19–21 . 10.1080/14786446208643307 .
• Good . Michael R.R. . Anderson . Paul R. . Evans . Charles R. . 2013 . Time dependence of particle creation from accelerating mirrors . Physical Review D . 88 . 2 . 025023 . 1303.6756 . 10.1103/PhysRevD.88.025023 .
• Web site: Gottschalk . Walter . Walter_Gottschalk . 2003 . Good Things about the Gudermannian . Gottschalk's Gestalts .
• Book: Gudermann, Christoph . Christoph Gudermann . 1833 . Theorie der Potenzial- oder cyklisch-hyperbolischen Functionen . G. Reimer . de . Theory of Potential- or Circular-Hyperbolic Functions .
• Jennings . George . Ni . David . Pong . Wai Yan . Raianu . Serban . 2022 . The Integral of Secant and Stereographic Projections of Conic Sections . math.HO . 2204.11187 .
• Kahlig . Peter . Reich . Ludwig . 2013 . Contributions to the theory of the Legendre-Gudermann equation . Fachbibliothek für Mathematik, Karl-Franzens-Universität Graz .
• Karney . Charles F.F. . 2011 . Transverse Mercator with an accuracy of a few nanometers . Journal of Geodesy . 85 . 8 . 475–485 . 1002.1417 . 10.1007/s00190-011-0445-3 .
• Kennelly . Arthur E. . 1928 . Gudermannian Complex Angles . Proceedings of the National Academy of Sciences . 14 . 11 . 839–844 . 10.1073/pnas.14.11.839 . free . 1085762 .
• Kennelly . Arthur E. . 1929 . Gudermannians and Lambertians with Their Respective Addition Theorems . Proceedings of the American Philosophical Society . 68 . 3 . 175–184 .
• Lambert . Johann Heinrich . Johann Heinrich Lambert . 1761 . 1768 . Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques . fr . Memoir on some remarkable properties of the circular and logarithmic transcendental quantities . Histoire de l'Académie Royale des Sciences et des Belles-Lettres . Berlin . 17 . 265–322 .
• Book: Lee, Laurence Patrick . Laurence Patrick Lee . 1976 . Conformal Projections Based on Elliptic Functions . Toronto . B. V. Gutsell, York University . Cartographica Monographs . 16 . limited . 0-919870-16-3 . Supplement No. 1 to The Canadian Cartographer 13.
• Book: Legendre, Adrien-Marie . Adrien-Marie Legendre . 1817 . Exercices de calcul intégral . 2 . Exercises in integral calculus . fr . Courcier .
• Majernik . V. . 1986 . Representation of relativistic quantities by trigonometric functions . American Journal of Physics . 54 . 6 . 536–538 . 10.1119/1.14557 .
• Book: McMahon, James . James McMahon (mathematician) . 1906 . Hyperbolic Functions . Wiley . [First published as {{cite book |last=McMahon |editor1-last=Merriman |editor2-last=Woodward |year=1896 |publisher=Wiley |chapter=IV. Hyperbolic Functions |title=Higher Mathematics |chapter-url=https://archive.org/details/highermathematic00merrrich/page/107 |pages=107–168 }}]
• Web site: Masson . Paul . The Complex Gudermannian . 2021 . Analytic Physics.
• Web site: Osborne . Peter . 2013 . The Mercator projections .
• Peters . J. M. H. . 1984 . The Gudermannian . The Mathematical Gazette . 68 . 445 . 192–196 . 10.2307/3616342 . 3616342 .
• Reynolds . William F. . 1993 . Hyperbolic Geometry on a Hyperboloid . The American Mathematical Monthly . 100 . 5 . 442–455 . 10.1080/00029890.1993.11990430 . https://web.archive.org/web/20160528134406id_/http://people.ucsc.edu:80/~rmont/classes/ClassicalGeometry/web2016/sources/Hyperbolic_AmerMathMonthly.pdf . 2016-05-28 . dead .
• Rickey . V. Frederick . Tuchinsky . Philip M. . 1980 . An application of geography to mathematics: History of the integral of the secant . Mathematics Magazine . 53 . 3 . 162–166 . 10.1080/0025570X.1980.11976846 .
• Ringermacher . Harry I. . Mead . Lawrence R. . 2009 . A new formula describing the scaffold structure of spiral galaxies . Monthly Notices of the Royal Astronomical Society . 397 . 1 . 164–171 . 10.1111/j.1365-2966.2009.14950.x . free . 0908.0892 .
• Robertson . John S. . 1997 . Gudermann and the simple pendulum . The College Mathematics Journal . 28 . 4 . 271–276 . 10.2307/2687148 . 2687148 .
• Romakina . Lyudmila N. . 2018 . The inverse Gudermannian in the hyperbolic geometry . Integral Transforms and Special Functions . 29 . 5 . 384–401 . 10.1080/10652469.2018.1441296 .
  • Sala . Kenneth L. . 1989 . Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean . SIAM Journal on Mathematical Analysis . 20 . 6 . 1514–1528 . 10.1137/0520100 .

External links

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

Notes and References

1. The symbols $\backslash psi$ and $\backslash phi$ 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, .
3. §4.23(viii) "Gudermannian Function";
4. pp. 23–27
5. draws complex-valued plots of several of these, demonstrating that naïve implementations that choose the principal branch of inverse trigonometric functions yield incorrect results.
6. p. 181; p. 269
7. p. 269 – note the typo.
8. §4.2.8(163) pp. 144–145
9. p. 182
10. p. 21
11. pp. 180–183
12. §4.2.7(162) pp. 143–144
13. Book: Turnbull . Herbert Westren . 1939 . James Gregory; Tercentenary Memorial Volume . G. Bell & Sons . 170 .
14. For example Hoüel labels the hyperbolic functions across the top in Table XIV of: Book: Hoüel, Guillaume Jules . 1885 . Recueil de formules et de tables numériques . Gauthier-Villars . 36 .
15. p. 74