Schläfli Function

The function giving the Volume of the spherical quadrectangular Tetrahedron:

$\begin{displaymath} V={\pi^2\over 8} f\left({{\pi \over p}, {\pi\over q}, {\pi\over r}}\right), \end{displaymath}$

where

${\textstyle{1\over 2}}\pi^2 f\left({{\pi\over 2}-x, y, {\pi\over 2}-z}\right)$

$= \sum_{m=1}^\infty \left({D-\sin x\sin z\over D+\sin x\sin z}\right)^m {\cos(2mx)-\cos(2my)+\cos(2mz)-1\over m^2}-x^2-y^2-z^2,$

and

$\begin{displaymath} D\equiv \sqrt{\cos^2 x\cos^2 z-\cos^2 y}. \end{displaymath}$

See also Tetrahedron

${\textstyle{1\over 2}}\pi^2 f\left({{\pi\over 2}-x, y, {\pi\over 2}-z}\right)$
$= \sum_{m=1}^\infty \left({D-\sin x\sin z\over D+\sin x\sin z}\right)^m {\cos(2mx)-\cos(2my)+\cos(2mz)-1\over m^2}-x^2-y^2-z^2,$