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Gudermannian function

The Gudermannian function relates the circular and hyperbolic trigonometric functions without resorting to complex numbers. It is defined by

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

Note that

\tanh\frac{x}{2} = \tan \frac{\mbox{gd}(x)}{2}.

The following identities also hold:

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

The inverse Gudermannian function is given by

{\rm gd}^{-1}(x)=\int_0^x \frac{dt}{\cos t}=\ln(\tan x+\sec x).\,

The derivatives of the Gudermannian and its inverse are

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

See also

References

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

External links

Science Fair Project Encyclopedia Contents Page
10-26-2009 08:16:03
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