convert/Elliptic_related converts - when possible - special functions admitting a 2F1 hypergeometric representation into Elliptic related functions; that is, into any of:

FunctionAdvisor( Elliptic_related );

The 11 functions in the "Elliptic_related" class are:

$\left[\mathrm{EllipticCE}\,\mathrm{EllipticCK}\,\mathrm{EllipticCPi}\,\mathrm{EllipticE}\,\mathrm{EllipticF}\,\mathrm{EllipticK}\,\mathrm{EllipticModulus}\,\mathrm{EllipticNome}\,\mathrm{EllipticPi}\,\mathrm{InverseJacobiAM}\,\mathrm{InverseJacobiSN}\right]$

$\mathrm{LegendreP}\left(\frac{1}{2}\,z\right)$

$\mathrm{LegendreP}\left(\frac{1}{2}\,z\right)$

$\mathrm{convert}\left(\,\mathrm{Elliptic\_related}\right)$

$\frac{2\sqrt{2}\sqrt{z+1}\mathrm{EllipticE}\left(\sqrt{\frac{z-1}{z+1}}\right)}{\mathrm{\pi }}-\frac{2\sqrt{2}\mathrm{EllipticK}\left(\sqrt{\frac{z-1}{z+1}}\right)}{\mathrm{\pi }\sqrt{z+1}}$

$\mathrm{hypergeom}\left(\left[-\frac{1}{2}\,\frac{1}{2}\right]\,\left[1\right]\,z^2\right)$

$\mathrm{hypergeom}\left(\left[-\frac{1}{2}\,\frac{1}{2}\right]\,\left[1\right]\,z^2\right)$

$\mathrm{convert}\left(\,\mathrm{Elliptic\_related}\right)$

$\frac{2\mathrm{EllipticE}\left(z\right)}{\mathrm{\pi }}$

$\frac{1}{2}\mathrm{\pi }\mathrm{hypergeom}\left(\left[\frac{1}{2}\,\frac{1}{2}\right]\,\left[1\right]\,1-z^2\right)$

$\frac{\mathrm{\pi }\mathrm{hypergeom}\left(\left[\frac{1}{2}\,\frac{1}{2}\right]\,\left[1\right]\,-z^2+1\right)}{2}$

$\mathrm{convert}\left(\,\mathrm{Elliptic\_related}\right)$

$\mathrm{EllipticCK}\left(z\right)$

$\frac{1}{4}\mathrm{\pi }{z}^2\left(\mathrm{LegendreP}\left(-\frac{1}{2}\,-1+2z^2\right)+\mathrm{LegendreP}\left(\frac{1}{2}\,-\frac{z^2-2}{z^2}\right)\mathrm{sqrt}\left(\frac{1}{z^2}\right)\right)$

$\frac{\mathrm{\pi }{z}^2\left(\mathrm{LegendreP}\left(-\frac{1}{2}\,2z^2-1\right)+\mathrm{LegendreP}\left(\frac{1}{2}\,-\frac{z^2-2}{z^2}\right)\sqrt{\frac{1}{z^2}}\right)}{4}$

$\mathrm{convert}\left(\,\mathrm{Elliptic\_related}\right)$

$\mathrm{EllipticCE}\left(z\right)$

$\mathrm{LegendreP}\left(-\frac{1}{2}\,1-2z^2\right)$

$\mathrm{LegendreP}\left(-\frac{1}{2}\,-2z^2+1\right)$

$\mathrm{convert}\left(\,\mathrm{Elliptic\_related}\right)$

$\frac{2\mathrm{EllipticK}\left(z\right)}{\mathrm{\pi }}$

$\mathrm{MeijerG}\left(\left[\left[\frac{1}{2}\,\frac{3}{2}\right]\,\left[\right]\right]\,\left[\left[0\right]\,\left[0\right]\right]\,-1+z^2\right)$

$\mathrm{MeijerG}\left(\left[\left[\frac{1}{2}\,\frac{3}{2}\right]\,\left[\right]\right]\,\left[\left[0\right]\,\left[0\right]\right]\,z^2-1\right)$

$\mathrm{convert}\left(\,\mathrm{Elliptic\_related}\right)$

$-4\mathrm{EllipticCE}\left(z\right)$