Gudermannian function

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Gudermannian.png
Gudermannian function with its asymptotes y = ±π/2 marked in gray.

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)=\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>{\rm gd}^{-1}(x)=\int_0^x \frac{dt}{\cos t}\,<math>
<math>=\ln(\tan x+\sec x)\,<math>
<math>=\ln \tan \left(\frac{\pi}{4} + \frac{x}{2}\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}\,\mbox{gd}^{-1}(x)=\sec(x)<math>

See also

References

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

External links

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