Legendre - Maple Help

convert/Legendre

convert special functions admitting 2F1 hypergeometric representation into Legendre functions

 Calling Sequence convert(expr, Legendre)

Parameters

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

Description

 • convert/Legendre converts, when possible, special functions admitting a 2F1 hypergeometric representation into Legendre functions. The Legendre functions are
 The 2 functions in the "Legendre" class are:
 $\left[{\mathrm{LegendreP}}{,}{\mathrm{LegendreQ}}\right]$ (1)

Examples

 > $\mathrm{arccoth}\left(z\right)$
 ${\mathrm{arccoth}}{}\left({z}\right)$ (2)
 > $\mathrm{convert}\left(,\mathrm{Legendre}\right)$
 ${\mathrm{LegendreQ}}{}\left({0}{,}\frac{{1}}{{z}}\right){+}\frac{{\mathrm{\pi }}{}\sqrt{{-}{\left({z}{-}{1}\right)}^{{2}}}}{{2}{}\left({z}{-}{1}\right)}$ (3)
 > $\frac{\frac{{\left(z+1\right)}^{\frac{1}{2}b}}{{\left(z-1\right)}^{\frac{1}{2}b}}\mathrm{\Gamma }\left(a+1\right)}{\mathrm{\Gamma }\left(1-b+a\right)}\mathrm{JacobiP}\left(a,-b,b,z\right)$
 $\frac{{\left({z}{+}{1}\right)}^{\frac{{b}}{{2}}}{}{\mathrm{\Gamma }}{}\left({a}{+}{1}\right){}{\mathrm{JacobiP}}{}\left({a}{,}{-}{b}{,}{b}{,}{z}\right)}{{\left({z}{-}{1}\right)}^{\frac{{b}}{{2}}}{}{\mathrm{\Gamma }}{}\left({1}{-}{b}{+}{a}\right)}$ (4)
 > $\mathrm{convert}\left(,\mathrm{Legendre}\right)$
 $\frac{{\mathrm{\Gamma }}{}\left({a}{+}{1}\right){}\left(\genfrac{}{}{0}{}{{a}{-}{b}}{{-}{b}}\right){}{\mathrm{\Gamma }}{}\left({-}{b}{+}{1}\right){}{\mathrm{LegendreP}}{}\left({a}{,}{b}{,}{z}\right)}{{\mathrm{\Gamma }}{}\left({1}{-}{b}{+}{a}\right)}$ (5)
 > $\mathrm{MeijerG}\left(\left[\left[-\frac{1}{2}a-\frac{1}{2}b,\frac{1}{2}-\frac{1}{2}a-\frac{1}{2}b\right],\left[\right]\right],\left[\left[0\right],\left[-\frac{1}{2}-a\right]\right],-\frac{1}{{z}^{2}}\right)$
 ${\mathrm{MeijerG}}{}\left(\left[\left[{-}\frac{{a}}{{2}}{-}\frac{{b}}{{2}}{,}\frac{{1}}{{2}}{-}\frac{{a}}{{2}}{-}\frac{{b}}{{2}}\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[{-}\frac{{1}}{{2}}{-}{a}\right]\right]{,}{-}\frac{{1}}{{{z}}^{{2}}}\right)$ (6)
 > $\mathrm{convert}\left(,\mathrm{Legendre}\right)$
 $\frac{{2}{}{{z}}^{{1}{+}{a}{+}{b}}{}{\mathrm{LegendreQ}}{}\left({a}{,}{b}{,}{z}\right)}{{{2}}^{{b}}{}{\left({z}{-}{1}\right)}^{\frac{{b}}{{2}}}{}{\left({z}{+}{1}\right)}^{\frac{{b}}{{2}}}{}{{ⅇ}}^{{I}{}{b}{}{\mathrm{\pi }}}}$ (7)

When converting to a function class, for example, Legendre, it is possible to request additional conversion rules to be performed. For instance, compare these two different outputs:

 > $\mathrm{GegenbauerC}\left(a,\frac{1}{2},z\right)$
 ${\mathrm{GegenbauerC}}{}\left({a}{,}\frac{{1}}{{2}}{,}{z}\right)$ (8)
 > $\mathrm{convert}\left(,\mathrm{Legendre}\right)$
 ${\mathrm{LegendreP}}{}\left({a}{,}{z}\right)$ (9)
 > $\mathrm{convert}\left(,\mathrm{Legendre},"raise a"\right)$
 $\frac{\left({3}{+}{2}{}{a}\right){}{z}{}{\mathrm{LegendreP}}{}\left({a}{+}{1}{,}{z}\right)}{{a}{+}{1}}{-}\frac{\left({a}{+}{2}\right){}{\mathrm{LegendreP}}{}\left({a}{+}{2}{,}{z}\right)}{{a}{+}{1}}$ (10)