Elliptic related - Maple Help

convert/Elliptic_related

convert - when possible - special functions admitting a 2F1 hypergeometric representation into Elliptic related functions

 Calling Sequence convert(expr, Elliptic_related)

Parameters

 expr - Maple expression, equation, or a set or list of them

Description

 • convert/Elliptic_related converts - when possible - special functions admitting a 2F1 hypergeometric representation into Elliptic related functions; that is, into any of:
 The 11 functions in the "Elliptic_related" class are:
 $\left[{\mathrm{EllipticCE}}{,}{\mathrm{EllipticCK}}{,}{\mathrm{EllipticCPi}}{,}{\mathrm{EllipticE}}{,}{\mathrm{EllipticF}}{,}{\mathrm{EllipticK}}{,}{\mathrm{EllipticModulus}}{,}{\mathrm{EllipticNome}}{,}{\mathrm{EllipticPi}}{,}{\mathrm{InverseJacobiAM}}{,}{\mathrm{InverseJacobiSN}}\right]$ (1)

Examples

 > $\mathrm{LegendreP}\left(\frac{1}{2},z\right)$
 ${\mathrm{LegendreP}}{}\left(\frac{{1}}{{2}}{,}{z}\right)$ (2)
 > $\mathrm{convert}\left(,\mathrm{Elliptic_related}\right)$
 $\frac{{2}{}\sqrt{{2}}{}\sqrt{{z}{+}{1}}{}{\mathrm{EllipticE}}{}\left(\sqrt{\frac{{z}{-}{1}}{{z}{+}{1}}}\right)}{{\mathrm{\pi }}}{-}\frac{{2}{}\sqrt{{2}}{}{\mathrm{EllipticK}}{}\left(\sqrt{\frac{{z}{-}{1}}{{z}{+}{1}}}\right)}{{\mathrm{\pi }}{}\sqrt{{z}{+}{1}}}$ (3)
 > $\mathrm{hypergeom}\left(\left[-\frac{1}{2},\frac{1}{2}\right],\left[1\right],{z}^{2}\right)$
 ${\mathrm{hypergeom}}{}\left(\left[{-}\frac{{1}}{{2}}{,}\frac{{1}}{{2}}\right]{,}\left[{1}\right]{,}{{z}}^{{2}}\right)$ (4)
 > $\mathrm{convert}\left(,\mathrm{Elliptic_related}\right)$
 $\frac{{2}{}{\mathrm{EllipticE}}{}\left({z}\right)}{{\mathrm{\pi }}}$ (5)
 > $\frac{1}{2}\mathrm{\pi }\mathrm{hypergeom}\left(\left[\frac{1}{2},\frac{1}{2}\right],\left[1\right],1-{z}^{2}\right)$
 $\frac{{\mathrm{\pi }}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{1}}{{2}}{,}\frac{{1}}{{2}}\right]{,}\left[{1}\right]{,}{-}{{z}}^{{2}}{+}{1}\right)}{{2}}$ (6)
 > $\mathrm{convert}\left(,\mathrm{Elliptic_related}\right)$
 ${\mathrm{EllipticCK}}{}\left({z}\right)$ (7)
 > $\frac{1}{4}\mathrm{\pi }{z}^{2}\left(\mathrm{LegendreP}\left(-\frac{1}{2},-1+2{z}^{2}\right)+\mathrm{LegendreP}\left(\frac{1}{2},-\frac{{z}^{2}-2}{{z}^{2}}\right)\mathrm{sqrt}\left(\frac{1}{{z}^{2}}\right)\right)$
 $\frac{{\mathrm{\pi }}{}{{z}}^{{2}}{}\left({\mathrm{LegendreP}}{}\left({-}\frac{{1}}{{2}}{,}{2}{}{{z}}^{{2}}{-}{1}\right){+}{\mathrm{LegendreP}}{}\left(\frac{{1}}{{2}}{,}{-}\frac{{{z}}^{{2}}{-}{2}}{{{z}}^{{2}}}\right){}\sqrt{\frac{{1}}{{{z}}^{{2}}}}\right)}{{4}}$ (8)
 > $\mathrm{convert}\left(,\mathrm{Elliptic_related}\right)$
 ${\mathrm{EllipticCE}}{}\left({z}\right)$ (9)
 > $\mathrm{LegendreP}\left(-\frac{1}{2},1-2{z}^{2}\right)$
 ${\mathrm{LegendreP}}{}\left({-}\frac{{1}}{{2}}{,}{-}{2}{}{{z}}^{{2}}{+}{1}\right)$ (10)
 > $\mathrm{convert}\left(,\mathrm{Elliptic_related}\right)$
 $\frac{{2}{}{\mathrm{EllipticK}}{}\left({z}\right)}{{\mathrm{\pi }}}$ (11)
 > $\mathrm{MeijerG}\left(\left[\left[\frac{1}{2},\frac{3}{2}\right],\left[\right]\right],\left[\left[0\right],\left[0\right]\right],-1+{z}^{2}\right)$
 ${\mathrm{MeijerG}}{}\left(\left[\left[\frac{{1}}{{2}}{,}\frac{{3}}{{2}}\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[{0}\right]\right]{,}{{z}}^{{2}}{-}{1}\right)$ (12)
 > $\mathrm{convert}\left(,\mathrm{Elliptic_related}\right)$
 ${-}{4}{}{\mathrm{EllipticCE}}{}\left({z}\right)$ (13)