Gudermannian function

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Image:Gudermannian.png The Gudermannian function, named after Christoph Gudermann (1798 - 1852), relates the circular and hyperbolic trigonometric functions without resorting to complex numbers. It is defined by

<math>{\rm gd}(x)\,</math>	<math>=\int_0^x \frac{dt}{\cosh t}</math>
	<math>=2\arctan \left(\tanh\frac{x}{2}\right)</math>
	<math>=2\arctan e^x-{\pi\over2}.</math>

Note that

<math>\tanh\frac{x}{2} = \tan \frac{\mbox{gd}(x)}{2}.\,</math>

The following identities also hold:

<math>\sinh(x)=\tan(\mbox{gd}(x))\ </math>

<math>\cosh(x)=\sec(\mbox{gd}(x))\ </math>

<math>\tanh(x)=\sin(\mbox{gd}(x))\ </math>

<math>\mbox{sech}(x)=\cos(\mbox{gd}(x))\ </math>

<math>\mbox{csch}(x)=\cot(\mbox{gd}(x))\ </math>

<math>\coth(x)=\csc(\mbox{gd}(x))\ </math>

The inverse Gudermannian function is given by

<math>\operatorname{arcgd}(x)</math>	<math>={\rm gd}^{-1}(x)=\int_0^x \frac{dt}{\cos t}\,</math>
	<math>=\operatorname{arccosh}(\sec x)=\operatorname{arctanh}(\sin x)\,</math>
	<math>=\ln\left(\sec(x)(1+\sin(x))\right)\,</math>
	<math>=\ln(\tan x+\sec x)=\ln\left(\tan\left(\frac{\pi}{4}+\frac{x}{2}\right)\right)\,</math>
	<math>=\frac{1}{2}\ln\left(\frac{1+\sin x}{1-\sin x} \right).\,</math>

The derivatives of the Gudermannian and its inverse are

<math>{d \over dx}\,\mbox{gd}(x)=\mbox{sech}(x),</math>

<math>{d \over dx}\,\operatorname{arcgd}(x)=\sec(x).</math>

See also

• hyperbolic secant distribution
• Mercator projection
• tangent half-angle formula
• tractrix
• trigonometric identity

References

• CRC Handbook of Mathematical Sciences 5th ed. pp 323-5.
• Template:Mathworld