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[[Image:Gudermannian.svg|thumb|270px|right|Gudermannian function with its [[asymptote]]s ''y''&nbsp;=&nbsp;±π/2 marked in blue]]
The '''Gudermannian function''', named after [[Christoph Gudermann]] (1798–1852), relates the [[circular function]]s and [[hyperbolic function]]s without using [[complex numbers]].

It is defined by

:<math>\begin{align}{\rm{gd}}\,x&=\int_0^x\frac{\mathrm{d}t}{\cosh t} \\[8pt]
&=\arcsin\left(\tanh x \right)

=\mathrm{arctan}\left(\sinh x \right) \\[8pt]
&=2\arctan\left[\tanh\left(\tfrac12x\right)\right]

=2\arctan(e^x)-\tfrac12\pi.
\end{align}\,\!</math>

Some related formulas don't quite work as definitions. For example, for real ''x'', <math>\arccos\mathrm{sech}\,x = \vert\mathrm{gd}\,x\vert = \arcsec(\cosh x)</math>. (See [[inverse trigonometric function]]s.)

The following identities hold:

:<math>\begin{align}{\color{white}\dot{{\color{black}
\sin\mathrm{gd}\,x}}}&=\tanh x ;\quad
\csc\mathrm{gd}\,x=\coth x ;\\
\cos\mathrm{gd}\,x&=\mathrm{sech}\, x ;\quad\,
\sec\mathrm{gd}\,x=\cosh x ;\\
\tan\mathrm{gd}\,x&=\sinh x ;\quad\,
\cot\mathrm{gd}\,x=\mathrm{csch}\, x ;\\
{}_{\color{white}.}\tan\tfrac{1}{2}\mathrm{gd}\,x&=\tanh\tfrac{1}{2}x.
\end{align}\,\!</math>

[[Image:GudermannianInverse.svg|thumb|270px|right|The inverse Gudermannian function]]

The [[inverse function|inverse]] Gudermannian function, which is defined on the interval &minus;''π''/2&nbsp;<&nbsp;''x''&nbsp;<&nbsp;''π''/2, is given by

:<math>
\begin{align}
\operatorname{gd}^{-1}\,x & = \int_0^x\frac{\mathrm{d}t}{\cos t} \\[8pt]
& = \ln\left| \frac{1 + \sin x}{\cos x} \right| = \tfrac12\ln \left| \frac{1 + \sin x}{1 - \sin x} \right| \\[8pt]
& = \ln\left| \tan x +\sec x \right| = \ln \left| \tan\left(\tfrac14\pi + \tfrac12x\right) \right| \\[8pt]
& = \mathrm{artanh}\,(\sin x) = \mathrm{arsinh}\,(\tan x).
\end{align}
</math>

(See [[inverse hyperbolic function]]s.)

The [[derivative]]s of the Gudermannian and its inverse are

:<math>\frac{\mathrm{d}}{\mathrm{d}x}\;\mathrm{gd}\,x=\mathrm{sech}\, x;
\quad \frac{\mathrm{d}}{\mathrm{d}x}\;\operatorname{gd}^{-1}\,x=\sec x.</math>

The expression 

:<math>\tfrac{1}{2}\pi - \mathrm{gd}\,x</math> 

defines the [[angle of parallelism]] function in [[hyperbolic geometry]].

==History==
The function 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 Gudermann's work in the 1830s on the theory of special functions.<ref>George F. Becker, C. E. Van Orstrand. ''Hyperbolic functions.'' Read Books, 1931. Page xlix.</ref> Gudermann had published articles in ''[[Crelle's Journal]]'' that were collected in ''Theorie der potenzial- oder cyklisch-hyperbolischen functionen'' (1833), a book which expounded ''sinh'' and ''cosh'' to a wide audience (under the guises of <math>\mathfrak{Sin}</math> and <math>\mathfrak{Cos}</math>).

The notation ''gd'' first appears on page 19 of the ''[[Philosophical Magazine]]'', vol. XXIV, where Cayley starts by calling ''gd. u'' the inverse of the [[integral of the secant function]]:

:<math>u = \int_0^\phi \sec t \,\mathrm{d}t = \ln\tan\left(\tfrac14\pi+\tfrac12\phi\right)</math>

and then derives "the definition" of the transcendent:

:<math>\operatorname{gd} \,u = i^{-1}\ln\tan\left(\tfrac14\pi+\tfrac12ui\right)</math>

observing immediately that it is a real function of ''u''.

==Applications==

The Gudermannian of the [[latitude|latitudinal]] (due North/South) distance from the [[equator]] on a [[Mercator projection]] is the [[meridian arc]] length, i.e. actual latitude on the globe.

The Gudermannian appears in a non-periodic solution of the [[inverted pendulum]].<ref>John S. Robertson, "Gudermann and the Simple Pendulum", ''The College Mathematics Journal'' '''28''':4:271–276 (September 1997) [http://www.jstor.org/stable/2687148 at JSTOR]</ref>

==See also==
*[[Hyperbolic secant distribution]]
*[[Mercator projection]]
*[[Tangent half-angle formula]]
*[[Tractrix]]
*[[Trigonometric identity]]

==Notes==
{{reflist}}

==References==
* CRC ''Handbook of Mathematical Sciences'' 5th&nbsp;ed. pp. 323–325.
* {{mathworld|urlname=Gudermannian|title=Gudermannian}}

[[Category:Trigonometry]]
[[Category:Elementary special functions]]
[[Category:Exponentials]]