Die Gudermannfunktion, benannt nach Christoph Gudermann (1798 - 1852), stellt eine Verbindung zwischen den trigonometrischen- und den hyperbolischen Funktionen her, ohne dabei die komplexen Zahlen zu benutzen.

Sie ist definiert durch:

$\{\backslash rm\; gd\}(x)\backslash ,$	$:=\backslash int\_0^x\; \backslash frac\{\backslash mathrm\; dt\}\{\backslash cosh\; t\}$
	$=2\backslash arctan\; \backslash left(\backslash tanh\backslash frac\{x\}\{2\}\backslash right)$

Für reelle x gilt sogar:

$\{\backslash rm\; gd\}(x)\backslash ,$	$=\backslash ,\backslash dots\; \backslash ,$
	$=\backslash arctan\; \backslash left(\backslash sinh\; x\backslash right)$
	$=2\backslash arctan\; \backslash left(e^x\backslash right)-\{\backslash pi\backslash over2\}.$

Die Umkehrfunktion der Gudermannfunktion ist definiert durch

$\backslash operatorname\{arcgd\}(x)$	$:=\{\backslash rm\; gd\}^\{-1\}(x)=\backslash int\_0^x\; \backslash frac\{\backslash mathrm\; dt\}\{\backslash cos\; t\}\backslash ,$
	$=\backslash operatorname\{arccosh\}(\backslash sec\; x)\backslash ,$
	$=\backslash operatorname\{arctanh\}(\backslash sin\; x)\backslash ,$
	$=\backslash ln\backslash left(\backslash sec(x)(1+\backslash sin(x))\backslash right)\backslash ,$
	$=\backslash ln(\backslash tan\; x+\backslash sec\; x)\backslash ,$
	$=\backslash ln\backslash left(\backslash tan\backslash left(\backslash frac\{\backslash pi\}\{4\}+\backslash frac\{x\}\{2\}\backslash right)\backslash right)\backslash ,$
	$=\backslash frac\{1\}\{2\}\backslash ln\backslash left(\backslash frac\{1+\backslash sin\; x\}\{1-\backslash sin\; x\}\; \backslash right).\backslash ,$

Die Ableitung der Gudermannfunktion und deren Umkehrung ist:

$\{\backslash mathrm\; d\; \backslash over\; \backslash mathrm\; dx\}\backslash ,\backslash mbox\{gd\}(x)=\backslash mbox\{sech\}(x),$

$\{\backslash mathrm\; d\; \backslash over\; \backslash mathrm\; dx\}\backslash ,\backslash operatorname\{arcgd\}(x)=\backslash sec(x).$

Besonders bemerkenswert sind die Identitäten:

$\backslash tanh\backslash frac\{x\}\{2\}\; =\; \backslash tan\; \backslash frac\{\backslash mbox\{gd\}(x)\}\{2\}.\backslash ,$

$\backslash operatorname\{gd\}\backslash left(\{\backslash imath\}\backslash cdot\; x\; \backslash right)\; =\; \{\backslash imath\}\; \backslash cdot\; \backslash operatorname\{arcgd\}\backslash left(x\backslash right)$

Die Verbindung von Kreis- und Hyperbelfunktionen ist gewährleistet durch:

$\backslash sinh(x)=\backslash tan(\backslash mbox\{gd\}(x))\backslash$

$\backslash cosh(x)=\backslash sec(\backslash mbox\{gd\}(x))\backslash$

$\backslash tanh(x)=\backslash sin(\backslash mbox\{gd\}(x))\backslash$

$\backslash mbox\{sech\}(x)=\backslash cos(\backslash mbox\{gd\}(x))\backslash$

$\backslash mbox\{csch\}(x)=\backslash cot(\backslash mbox\{gd\}(x))\backslash$

$\backslash coth(x)=\backslash csc(\backslash mbox\{gd\}(x))\backslash$

Daraus ergibt sich folgender Zusammenhang mit der Exponentialfunktion:

{| $\backslash exp\backslash left(x\backslash right)$ $=\backslash frac\{1\}\{\backslash cos\backslash left(\backslash operatorname\{gd\}\backslash left(x\backslash right)\backslash right)\}+\backslash tan\backslash left(\backslash operatorname\{gd\}\backslash left(x\backslash right)\backslash right)=\backslash sec\backslash left(\backslash operatorname\{gd\}\backslash left(x\backslash right)\backslash right)+\backslash tan\backslash left(\backslash operatorname\{gd\}\backslash left(x\backslash right)\backslash right)$ $=\backslash tan\backslash left(\backslash frac\{\backslash pi\}\{4\}+\backslash frac\{\backslash operatorname\{gd\}\backslash left(x\backslash right)\}\{2\}\backslash right)$ $=\backslash frac\{1+\backslash sin\backslash left(\backslash operatorname\{gd\}\backslash left(x\backslash right)\backslash right)\}\{\backslash cos\backslash left(\backslash operatorname\{gd\}\backslash left(x\backslash right)\backslash right)\}$

Praktische Anwendung

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Mit der Gudermannfunktion bzw deren Umkehrung kann man Latitude φ und Longitude λ der Mercator-Projektion berechnen.

Quellen

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• Eintrag über Gudermannfunktion bei MathWorld

Gudermannian function

Trigonometrie