Gudermannian function

The Gudermannian function, named after Christoph Gudermann (1798 – 1852), relates the circular and hyperbolic trigonometric functions without using complex numbers.

It is defined by

: $egin{align}{
m{gd(x)&=int_0^xfrac{dp}{cosh(p)}\&=arcsinleft( anh(x)
ight) =mbox{arccsc}left(coth(x)
ight)\&=arccosleft(mbox{sech}(x)
ight) =mbox{arcsec}left(cosh(x)
ight)\&=arctanleft(sinh(x)
ight) =mbox{arccot}left(mbox{csch}(x)
ight) \&=2arctanleft( anhleft(frac{x}{2}
ight)
ight)=2arctan(e^x)-frac{pi}{2}.end{align},!$

The following identities also hold:

: $egin{align}{color{white}dot$ color{black}sin(mbox{gd}(x))}&= anh(x);quadcsc(mbox{gd}(x))=coth(x);\cos(mbox{gd}(x))&=mbox{sech}(x);quad,sec(mbox{gd}(x))=cosh(x);\ an(mbox{gd}(x))&=sinh(x);quad,cot(mbox{gd}(x))=mbox{csch}(x);\{}_{color{white}.} anleft(frac{mbox{gd}(x)}{2} ight)&= anhleft(frac{x}{2} ight).end{align},!

The inverse Gudermannian function is given by

: $egin{align}mbox{arcgd}(x)&={
m {gd^{-1}(x)=int_0^xfrac{dp}{cos(p)}\&={}mbox{arccosh}(sec(x))=mbox{arctanh}(sin(x))\&={}lnleft(sec(x)(1+sin(x))
ight)\&={}ln( an(x)+sec(x))=ln anleft(frac{pi}{4}+frac{x}{2}
ight)\&={}frac{1}{2}ln frac{1+sin(x)}{1-sin(x)} .end{align},!$

The derivatives of the Gudermannian and its inverse are

: $frac{d}{dx};mbox{gd}(x)=mbox{sech}(x);quadfrac{d}{dx};mbox{arcgd}(x)=sec(x).,!$

ee also

*Hyperbolic secant distribution
*Mercator projection
*Tangent half-angle formula
*Tractrix
*Trigonometric identity

References

* CRC "Handbook of Mathematical Sciences" 5th ed. pp 323–5.
*

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