Heun - Maple Help

convert/Heun

convert to special functions of the Heun class

 Calling Sequence convert(expr, Heun)

Parameters

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

Description

 • convert/Heun converts, when possible, hypergeometric, MeijerG and special functions into Heun functions; that is, into one of
 The 23 functions in the "Heun" class are:
 $\left[{\mathrm{HeunB}}{,}{\mathrm{HeunBPrime}}{,}{\mathrm{HeunC}}{,}{\mathrm{HeunCPrime}}{,}{\mathrm{HeunD}}{,}{\mathrm{HeunDPrime}}{,}{\mathrm{HeunG}}{,}{\mathrm{HeunGPrime}}{,}{\mathrm{HeunT}}{,}{\mathrm{HeunTPrime}}{,}{\mathrm{MathieuA}}{,}{\mathrm{MathieuB}}{,}{\mathrm{MathieuC}}{,}{\mathrm{MathieuCE}}{,}{\mathrm{MathieuCEPrime}}{,}{\mathrm{MathieuCPrime}}{,}{\mathrm{MathieuExponent}}{,}{\mathrm{MathieuFloquet}}{,}{\mathrm{MathieuFloquetPrime}}{,}{\mathrm{MathieuS}}{,}{\mathrm{MathieuSE}}{,}{\mathrm{MathieuSEPrime}}{,}{\mathrm{MathieuSPrime}}\right]$ (1)
 • convert/Heun accepts as optional arguments all those described in convert[to_special_function].

Examples

An assorted sample of special and elementary functions

 > $\mathrm{functions_2F1}≔\left[\mathrm{ChebyshevT},\mathrm{JacobiP},\mathrm{SphericalY},\mathrm{EllipticK},\mathrm{GaussAGM},\mathrm{arctan},\mathrm{arcsin}\right]$
 ${\mathrm{functions_2F1}}{≔}\left[{\mathrm{ChebyshevT}}{,}{\mathrm{JacobiP}}{,}{\mathrm{SphericalY}}{,}{\mathrm{EllipticK}}{,}{\mathrm{GaussAGM}}{,}{\mathrm{arctan}}{,}{\mathrm{arcsin}}\right]$ (2)

Their syntax (calling sequence) in Maple

 > $\mathrm{map2}\left(\mathrm{FunctionAdvisor},\mathrm{syntax},\mathrm{functions_2F1}\right)$
 $\left[{\mathrm{ChebyshevT}}{}\left({a}{,}{z}\right){,}{\mathrm{JacobiP}}{}\left({a}{,}{b}{,}{c}{,}{z}\right){,}{\mathrm{SphericalY}}{}\left({\mathrm{\lambda }}{,}{\mathrm{\mu }}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){,}{\mathrm{EllipticK}}{}\left({k}\right){,}{\mathrm{GaussAGM}}{}\left({x}{,}{y}\right){,}{\mathrm{arctan}}{}\left({y}{,}{x}\right){,}{\mathrm{arcsin}}{}\left({z}\right)\right]$ (3)

A Heun representation for them, in these cases using HeunC

 > $\mathrm{map}\left(u↦u=\mathrm{convert}\left(u,\mathrm{Heun}\right),\right)$
 $\left[{\mathrm{ChebyshevT}}{}\left({a}{,}{z}\right){=}{\mathrm{HeunC}}{}\left({0}{,}{-}\frac{{1}}{{2}}{,}{-}{2}{}{a}{,}{0}{,}{{a}}^{{2}}{+}\frac{{1}}{{4}}{,}\frac{{z}{-}{1}}{{z}{+}{1}}\right){}{\left(\frac{{1}}{{2}}{+}\frac{{z}}{{2}}\right)}^{{a}}{,}{\mathrm{JacobiP}}{}\left({a}{,}{b}{,}{c}{,}{z}\right){=}\frac{\left(\genfrac{}{}{0}{}{{a}{+}{b}}{{b}}\right){}{\mathrm{HeunC}}{}\left({0}{,}{b}{,}{b}{+}{c}{+}{2}{}{a}{+}{1}{,}{0}{,}\frac{\left({b}{+}{1}{+}{2}{}{a}\right){}\left({b}{+}{c}{+}{a}{+}{1}\right)}{{2}}{-}\frac{{b}}{{2}}{-}\frac{\left({b}{+}{1}\right){}{a}}{{2}}{,}\frac{{z}{-}{1}}{{z}{+}{1}}\right)}{{\left(\frac{{1}}{{2}}{+}\frac{{z}}{{2}}\right)}^{{b}{+}{c}{+}{a}{+}{1}}}{,}{\mathrm{SphericalY}}{}\left({\mathrm{\lambda }}{,}{\mathrm{\mu }}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){=}\frac{{\left({-1}\right)}^{{\mathrm{\mu }}}{}\sqrt{\frac{{2}{}{\mathrm{\lambda }}{+}{1}}{{\mathrm{\pi }}}}{}\sqrt{\left({\mathrm{\lambda }}{-}{\mathrm{\mu }}\right){!}}{}{{ⅇ}}^{{I}{}{\mathrm{\phi }}{}{\mathrm{\mu }}}{}{\left({\mathrm{cos}}{}\left({\mathrm{\theta }}\right){+}{1}\right)}^{\frac{{\mathrm{\mu }}}{{2}}}{}{\mathrm{HeunC}}{}\left({0}{,}{-}{\mathrm{\mu }}{,}{2}{}{\mathrm{\lambda }}{+}{1}{,}{0}{,}{{\mathrm{\lambda }}}^{{2}}{+}{\mathrm{\lambda }}{+}\frac{{1}}{{2}}{,}\frac{{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){-}{1}}{{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){+}{1}}\right)}{{2}{}\sqrt{\left({\mathrm{\lambda }}{+}{\mathrm{\mu }}\right){!}}{}{\left({\mathrm{cos}}{}\left({\mathrm{\theta }}\right){-}{1}\right)}^{\frac{{\mathrm{\mu }}}{{2}}}{}{\mathrm{\Gamma }}{}\left({1}{-}{\mathrm{\mu }}\right){}{\left(\frac{{1}}{{2}}{+}\frac{{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)}{{2}}\right)}^{{\mathrm{\lambda }}{+}{1}}}{,}{\mathrm{EllipticK}}{}\left({k}\right){=}\frac{{\mathrm{\pi }}{}{\mathrm{HeunC}}{}\left({0}{,}{0}{,}{0}{,}{0}{,}\frac{{1}}{{4}}{,}\frac{{{k}}^{{2}}}{{{k}}^{{2}}{-}{1}}\right)}{{2}{}\sqrt{{-}{{k}}^{{2}}{+}{1}}}{,}{\mathrm{GaussAGM}}{}\left({x}{,}{y}\right){=}\frac{\left({x}{+}{y}\right){}\sqrt{{4}}{}\sqrt{\frac{{x}{}{y}}{{\left({x}{+}{y}\right)}^{{2}}}}}{{2}{}{\mathrm{HeunC}}{}\left({0}{,}{0}{,}{0}{,}{0}{,}\frac{{1}}{{4}}{,}{-}\frac{{\left({x}{-}{y}\right)}^{{2}}}{{4}{}{y}{}{x}}\right)}{,}{\mathrm{arctan}}{}\left({y}{,}{x}\right){=}\frac{{\mathrm{HeunC}}{}\left({0}{,}{1}{,}{0}{,}{0}{,}\frac{{1}}{{2}}{,}\frac{{I}{}{y}{+}{x}{-}\sqrt{{{x}}^{{2}}{+}{{y}}^{{2}}}}{\sqrt{{{x}}^{{2}}{+}{{y}}^{{2}}}{}\left({1}{+}\frac{{I}{}{y}{+}{x}{-}\sqrt{{{x}}^{{2}}{+}{{y}}^{{2}}}}{\sqrt{{{x}}^{{2}}{+}{{y}}^{{2}}}}\right)}\right){}\left({y}{+}{I}{}\sqrt{{{x}}^{{2}}{+}{{y}}^{{2}}}{-}{I}{}{x}\right)}{{x}{+}{I}{}{y}}{,}{\mathrm{arcsin}}{}\left({z}\right){=}\frac{{z}{}{\mathrm{HeunC}}{}\left({0}{,}\frac{{1}}{{2}}{,}{0}{,}{0}{,}\frac{{1}}{{4}}{,}\frac{{{z}}^{{2}}}{{{z}}^{{2}}{-}{1}}\right)}{\sqrt{{-}{{z}}^{{2}}{+}{1}}}\right]$ (4)

A sample of special and elementary functions not admitting HeunG representation

 > $\mathrm{functions_1F1}≔\left[\mathrm{erf}\left(z\right),\mathrm{dawson}\left(z\right),\mathrm{Ei}\left(a,z\right),\mathrm{LaguerreL}\left(a,b,z\right),\mathrm{hypergeom}\left(\left[a\right],\left[b\right],z\right),\mathrm{MeijerG}\left(\left[\left[a\right],\left[\right]\right],\left[\left[0\right],\left[b\right]\right],z\right),\mathrm{cos}\left(z\right),\mathrm{sin}\left(z\right)\right]$
 ${\mathrm{functions_1F1}}{≔}\left[{\mathrm{erf}}{}\left({z}\right){,}{\mathrm{dawson}}{}\left({z}\right){,}{{\mathrm{Ei}}}_{{a}}{}\left({z}\right){,}{\mathrm{LaguerreL}}{}\left({a}{,}{b}{,}{z}\right){,}{\mathrm{hypergeom}}{}\left(\left[{a}\right]{,}\left[{b}\right]{,}{z}\right){,}{\mathrm{MeijerG}}{}\left(\left[\left[{a}\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[{b}\right]\right]{,}{z}\right){,}{\mathrm{cos}}{}\left({z}\right){,}{\mathrm{sin}}{}\left({z}\right)\right]$ (5)

By default, the results are returned in terms of the lower Heun functions, that is, those with less parameters, in this case HeunB

 > $\mathrm{map}\left(u↦u=\mathrm{convert}\left(u,\mathrm{Heun}\right),\mathrm{functions_1F1}\right)$
 $\left[{\mathrm{erf}}{}\left({z}\right){=}\frac{{2}{}{z}{}{\mathrm{HeunB}}{}\left({1}{,}{0}{,}{1}{,}{0}{,}\sqrt{{-}{{z}}^{{2}}}\right)}{\sqrt{{\mathrm{\pi }}}}{,}{\mathrm{dawson}}{}\left({z}\right){=}\frac{{z}{}{\mathrm{HeunB}}{}\left({1}{,}{0}{,}{1}{,}{0}{,}\sqrt{{{z}}^{{2}}}\right)}{{{ⅇ}}^{{{z}}^{{2}}}}{,}{{\mathrm{Ei}}}_{{a}}{}\left({z}\right){=}\frac{{\mathrm{HeunB}}{}\left({2}{-}{2}{}{a}{,}{0}{,}{2}{}{a}{,}{0}{,}\sqrt{{-}{z}}\right)}{{a}{-}{1}}{+}{{z}}^{{a}{-}{1}}{}{\mathrm{\Gamma }}{}\left({1}{-}{a}\right){,}{\mathrm{LaguerreL}}{}\left({a}{,}{b}{,}{z}\right){=}\left(\genfrac{}{}{0}{}{{a}{+}{b}}{{a}}\right){}{\mathrm{HeunB}}{}\left({2}{}{b}{,}{0}{,}{2}{}{b}{+}{2}{+}{4}{}{a}{,}{0}{,}\sqrt{{z}}\right){,}{\mathrm{hypergeom}}{}\left(\left[{a}\right]{,}\left[{b}\right]{,}{z}\right){=}{\mathrm{HeunB}}{}\left({2}{}{b}{-}{2}{,}{0}{,}{2}{}{b}{-}{4}{}{a}{,}{0}{,}\sqrt{{z}}\right){,}{\mathrm{MeijerG}}{}\left(\left[\left[{a}\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[{b}\right]\right]{,}{z}\right){=}\frac{{\mathrm{\Gamma }}{}\left({1}{-}{a}\right){}{\mathrm{HeunB}}{}\left({-}{2}{}{b}{,}{0}{,}{-}{2}{-}{2}{}{b}{+}{4}{}{a}{,}{0}{,}\sqrt{{-}{z}}\right)}{{\mathrm{\Gamma }}{}\left({1}{-}{b}\right)}{,}{\mathrm{cos}}{}\left({z}\right){=}\frac{\left({2}{}{z}{+}{\mathrm{\pi }}\right){}{\mathrm{HeunB}}{}\left({2}{,}{0}{,}{0}{,}{0}{,}\sqrt{{I}{}\left({2}{}{z}{+}{\mathrm{\pi }}\right)}\right)}{{2}{}{{ⅇ}}^{\frac{{I}}{{2}}{}\left({2}{}{z}{+}{\mathrm{\pi }}\right)}}{,}{\mathrm{sin}}{}\left({z}\right){=}\frac{{z}{}{\mathrm{HeunB}}{}\left({2}{,}{0}{,}{0}{,}{0}{,}\sqrt{{2}}{}\sqrt{{I}{}{z}}\right)}{{{ⅇ}}^{{I}{}{z}}}\right]$ (6)

A representation in terms of higher Heun functions, in this case HeunC, because these functions being converted belong to the 1F1 class, can be obtained specifying HeunC instead of Heun in the call to convert

 > $\mathrm{map}\left(u↦u=\mathrm{convert}\left(u,\mathrm{HeunC}\right),\mathrm{functions_1F1}\right)$
 $\left[{\mathrm{erf}}{}\left({z}\right){=}\frac{\left({-}{2}{}{{z}}^{{3}}{+}{2}{}{z}\right){}{\mathrm{HeunC}}{}\left({1}{,}\frac{{1}}{{2}}{,}{1}{,}{-}\frac{{1}}{{4}}{,}\frac{{3}}{{4}}{,}{{z}}^{{2}}\right)}{\sqrt{{\mathrm{\pi }}}}{,}{\mathrm{dawson}}{}\left({z}\right){=}\frac{{z}{}{\mathrm{HeunC}}{}\left({1}{,}\frac{{1}}{{2}}{,}{1}{,}{-}\frac{{1}}{{4}}{,}\frac{{3}}{{4}}{,}{-}{{z}}^{{2}}\right){}\left({{z}}^{{2}}{+}{1}\right)}{{{ⅇ}}^{{{z}}^{{2}}}}{,}{{\mathrm{Ei}}}_{{a}}{}\left({z}\right){=}\frac{\left({1}{-}{z}\right){}{\mathrm{HeunC}}{}\left({1}{,}{1}{-}{a}{,}{1}{,}{-}\frac{{a}}{{2}}{,}\frac{{1}}{{2}}{+}\frac{{a}}{{2}}{,}{z}\right)}{{a}{-}{1}}{+}{{z}}^{{a}{-}{1}}{}{\mathrm{\Gamma }}{}\left({1}{-}{a}\right){,}{\mathrm{LaguerreL}}{}\left({a}{,}{b}{,}{z}\right){=}\left(\genfrac{}{}{0}{}{{a}{+}{b}}{{a}}\right){}{\mathrm{HeunC}}{}\left({1}{,}{b}{,}{1}{,}{-}\frac{{b}}{{2}}{-}\frac{{1}}{{2}}{-}{a}{,}\frac{{b}}{{2}}{+}{1}{+}{a}{,}{-}{z}\right){}\left({z}{+}{1}\right){,}{\mathrm{hypergeom}}{}\left(\left[{a}\right]{,}\left[{b}\right]{,}{z}\right){=}{\mathrm{HeunC}}{}\left({1}{,}{b}{-}{1}{,}{1}{,}{-}\frac{{b}}{{2}}{+}{a}{,}\frac{{b}}{{2}}{-}{a}{+}\frac{{1}}{{2}}{,}{-}{z}\right){}\left({z}{+}{1}\right){,}{\mathrm{MeijerG}}{}\left(\left[\left[{a}\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[{b}\right]\right]{,}{z}\right){=}\frac{{\mathrm{\Gamma }}{}\left({1}{-}{a}\right){}{\mathrm{HeunC}}{}\left({1}{,}{-}{b}{,}{1}{,}\frac{{b}}{{2}}{-}{a}{+}\frac{{1}}{{2}}{,}{-}\frac{{b}}{{2}}{+}{a}{,}{z}\right){}\left({1}{-}{z}\right)}{{\mathrm{\Gamma }}{}\left({1}{-}{b}\right)}{,}{\mathrm{cos}}{}\left({z}\right){=}\frac{\left({2}{}{z}{+}{\mathrm{\pi }}\right){}{\mathrm{HeunC}}{}\left({1}{,}{1}{,}{1}{,}{0}{,}\frac{{1}}{{2}}{,}{-I}{}\left({2}{}{z}{+}{\mathrm{\pi }}\right)\right){}\left({I}{}\left({2}{}{z}{+}{\mathrm{\pi }}\right){+}{1}\right)}{{2}{}{{ⅇ}}^{\frac{{I}}{{2}}{}\left({2}{}{z}{+}{\mathrm{\pi }}\right)}}{,}{\mathrm{sin}}{}\left({z}\right){=}\frac{\left({2}{}{I}{}{{z}}^{{2}}{+}{z}\right){}{\mathrm{HeunC}}{}\left({1}{,}{1}{,}{1}{,}{0}{,}\frac{{1}}{{2}}{,}{-}{2}{}{I}{}{z}\right)}{{{ⅇ}}^{{I}{}{z}}}\right]$ (7)