# 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

[itex]{\rm gd}(x)=\int_0^x \frac{dt}{\cosh t}[itex]
[itex]{}=2\arctan \left(\tanh\frac{x}{2}\right)[itex]
[itex]{}=2\arctan e^x-{\pi\over2}.[itex]

Note that

[itex]\tanh\frac{x}{2} = \tan \frac{\mbox{gd}(x)}{2}.\,[itex]

The following identities also hold:

[itex]\sinh(x)=\tan(\mbox{gd}(x))\ [itex]
[itex]\cosh(x)=\sec(\mbox{gd}(x))\ [itex]
[itex]\tanh(x)=\sin(\mbox{gd}(x))\ [itex]
[itex]\mbox{sech}(x)=\cos(\mbox{gd}(x))\ [itex]
[itex]\mbox{csch}(x)=\cot(\mbox{gd}(x))\ [itex]
[itex]\coth(x)=\csc(\mbox{gd}(x))\ [itex]

The inverse Gudermannian function is given by

[itex]{\rm gd}^{-1}(x)=\int_0^x \frac{dt}{\cos t}\,[itex]
[itex]=\ln(\tan x+\sec x)\,[itex]
[itex]=\ln \tan \left(\frac{\pi}{4} + \frac{x}{2}\right)\,[itex]
[itex]=\frac{1}{2}\ln\left(\frac{1+\sin x}{1-\sin x} \right)\,[itex]

The derivatives of the Gudermannian and its inverse are

[itex]{d \over dx}\,\mbox{gd}(x)=\mbox{sech}(x)[itex]
[itex]{d \over dx}\,\mbox{gd}^{-1}(x)=\sec(x)[itex]

## References

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

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