convert/Heun - Maple Programming Help

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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

 > $\mathrm{map2}\left(\mathrm{FunctionAdvisor},\mathrm{syntax},\mathrm{functions_2F1}\right)$
 $\left[{\mathrm{ChebyshevT}}{}\left({a}{,}{z}\right){,}{\mathrm{JacobiP}}{}\left({a}{,}{b}{,}{c}{,}{z}\right){,}{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){,}{\mathrm{EllipticK}}{}\left({k}\right){,}{\mathrm{GaussAGM}}{}\left({x}{,}{y}\right){,}{\mathrm{arctan}}{}\left({y}{,}{x}\right){,}{\mathrm{arcsin}}{}\left({z}\right)\right]$ (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{{1}}{{2}}{}{z}\right)}^{{a}}{,}{\mathrm{JacobiP}}{}\left({a}{,}{b}{,}{c}{,}{z}\right){=}\frac{{\mathrm{binomial}}{}\left({a}{+}{b}{,}{b}\right){}{\mathrm{HeunC}}{}\left({0}{,}{b}{,}{b}{+}{c}{+}{2}{}{a}{+}{1}{,}{0}{,}\frac{{1}}{{2}}{}\left({b}{+}{1}{+}{2}{}{a}\right){}\left({b}{+}{c}{+}{a}{+}{1}\right){-}\frac{{1}}{{2}}{}{b}{-}\frac{{1}}{{2}}{}{a}{}\left({b}{+}{1}\right){,}\frac{{z}{-}{1}}{{z}{+}{1}}\right)}{{\left(\frac{{1}}{{2}}{+}\frac{{1}}{{2}}{}{z}\right)}^{{b}{+}{c}{+}{a}{+}{1}}}{,}{\mathrm{SphericalY}}{}\left({\mathrm{λ}}{,}{\mathrm{μ}}{,}{\mathrm{θ}}{,}{\mathrm{φ}}\right){=}\frac{{1}}{{2}}{}\frac{{\left({-}{1}\right)}^{{\mathrm{μ}}}{}\sqrt{\frac{{2}{}{\mathrm{λ}}{+}{1}}{{\mathrm{π}}}}{}\sqrt{\left({\mathrm{λ}}{-}{\mathrm{μ}}\right){!}}{}{{ⅇ}}^{{I}{}{\mathrm{φ}}{}{\mathrm{μ}}}{}{\left({\mathrm{cos}}{}\left({\mathrm{θ}}\right){+}{1}\right)}^{\frac{{1}}{{2}}{}{\mathrm{μ}}}{}{\mathrm{HeunC}}{}\left({0}{,}{-}{\mathrm{μ}}{,}{2}{}{\mathrm{λ}}{+}{1}{,}{0}{,}{{\mathrm{λ}}}^{{2}}{+}{\mathrm{λ}}{+}\frac{{1}}{{2}}{,}\frac{{\mathrm{cos}}{}\left({\mathrm{θ}}\right){-}{1}}{{\mathrm{cos}}{}\left({\mathrm{θ}}\right){+}{1}}\right)}{\sqrt{\left({\mathrm{λ}}{+}{\mathrm{μ}}\right){!}}{}{\left({\mathrm{cos}}{}\left({\mathrm{θ}}\right){-}{1}\right)}^{\frac{{1}}{{2}}{}{\mathrm{μ}}}{}{\mathrm{Γ}}{}\left({1}{-}{\mathrm{μ}}\right){}{\left(\frac{{1}}{{2}}{+}\frac{{1}}{{2}}{}{\mathrm{cos}}{}\left({\mathrm{θ}}\right)\right)}^{{\mathrm{λ}}{+}{1}}}{,}{\mathrm{EllipticK}}{}\left({k}\right){=}\frac{{1}}{{2}}{}\frac{{\mathrm{π}}{}{\mathrm{HeunC}}{}\left({0}{,}{0}{,}{0}{,}{0}{,}\frac{{1}}{{4}}{,}\frac{{{k}}^{{2}}}{{{k}}^{{2}}{-}{1}}\right)}{\sqrt{{-}{{k}}^{{2}}{+}{1}}}{,}{\mathrm{GaussAGM}}{}\left({x}{,}{y}\right){=}\frac{{1}}{{2}}{}\frac{\left({x}{+}{y}\right){}\sqrt{{4}}{}\sqrt{\frac{{x}{}{y}}{{\left({x}{+}{y}\right)}^{{2}}}}}{{\mathrm{HeunC}}{}\left({0}{,}{0}{,}{0}{,}{0}{,}\frac{{1}}{{4}}{,}{-}\frac{{1}}{{4}}{}\frac{{\left({x}{-}{y}\right)}^{{2}}}{{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

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

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

 > $\mathrm{map}\left(u→u=\mathrm{convert}\left(u,\mathrm{Heun}\right),\mathrm{functions_1F1}\right)$
 $\left[{\mathrm{erf}}{}\left({z}\right){=}\frac{{2}{}{z}{}{\mathrm{HeunB}}{}\left({1}{,}{0}{,}{1}{,}{0}{,}\sqrt{{-}{{z}}^{{2}}}\right)}{\sqrt{{\mathrm{π}}}}{,}{\mathrm{dawson}}{}\left({z}\right){=}\frac{{z}{}{\mathrm{HeunB}}{}\left({1}{,}{0}{,}{1}{,}{0}{,}\sqrt{{{z}}^{{2}}}\right)}{{{ⅇ}}^{{{z}}^{{2}}}}{,}{\mathrm{Ei}}{}\left({a}{,}{z}\right){=}\frac{{\mathrm{HeunB}}{}\left({2}{-}{2}{}{a}{,}{0}{,}{2}{}{a}{,}{0}{,}\sqrt{{-}{z}}\right)}{{a}{-}{1}}{+}{{z}}^{{a}{-}{1}}{}{\mathrm{Γ}}{}\left({1}{-}{a}\right){,}{\mathrm{LaguerreL}}{}\left({a}{,}{b}{,}{z}\right){=}{\mathrm{binomial}}{}\left({a}{+}{b}{,}{a}\right){}{\mathrm{HeunB}}{}\left({2}{}{b}{,}{0}{,}{2}{}{b}{+}{2}{+}{4}{}{a}{,}{0}{,}\sqrt{{z}}\right){,}{\mathrm{hypergeom}}{}\left(\left[{a}\right]{,}\left[{b}\right]{,}{z}\right){=}{\mathrm{HeunB}}{}\left({2}{}{b}{-}{2}{,}{0}{,}{2}{}{b}{-}{4}{}{a}{,}{0}{,}\sqrt{{z}}\right){,}{\mathrm{MeijerG}}{}\left(\left[\left[{a}\right]{,}\left[{}\right]\right]{,}\left[\left[{0}\right]{,}\left[{b}\right]\right]{,}{z}\right){=}\frac{{\mathrm{Γ}}{}\left({1}{-}{a}\right){}{\mathrm{HeunB}}{}\left({-}{2}{}{b}{,}{0}{,}{-}{2}{-}{2}{}{b}{+}{4}{}{a}{,}{0}{,}\sqrt{{-}{z}}\right)}{{\mathrm{Γ}}{}\left({1}{-}{b}\right)}{,}{\mathrm{cos}}{}\left({z}\right){=}\frac{{1}}{{2}}{}\frac{\left({2}{}{z}{+}{\mathrm{π}}\right){}{\mathrm{HeunB}}{}\left({2}{,}{0}{,}{0}{,}{0}{,}\sqrt{{I}{}\left({2}{}{z}{+}{\mathrm{π}}\right)}\right)}{{{ⅇ}}^{\frac{{1}}{{2}}{}{I}{}\left({2}{}{z}{+}{\mathrm{π}}\right)}}{,}{\mathrm{sin}}{}\left({z}\right){=}\frac{{z}{}{\mathrm{HeunB}}{}\left({2}{,}{0}{,}{0}{,}{0}{,}\sqrt{{2}}{}\sqrt{{I}{}{z}}\right)}{{{ⅇ}}^{{I}{}{z}}}\right]$ (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{π}}}}{,}{\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}}{}\left({a}{,}{z}\right){=}\frac{\left({1}{-}{z}\right){}{\mathrm{HeunC}}{}\left({1}{,}{1}{-}{a}{,}{1}{,}{-}\frac{{1}}{{2}}{}{a}{,}\frac{{1}}{{2}}{+}\frac{{1}}{{2}}{}{a}{,}{z}\right)}{{a}{-}{1}}{+}{{z}}^{{a}{-}{1}}{}{\mathrm{Γ}}{}\left({1}{-}{a}\right){,}{\mathrm{LaguerreL}}{}\left({a}{,}{b}{,}{z}\right){=}{\mathrm{binomial}}{}\left({a}{+}{b}{,}{a}\right){}{\mathrm{HeunC}}{}\left({1}{,}{b}{,}{1}{,}{-}\frac{{1}}{{2}}{}{b}{-}\frac{{1}}{{2}}{-}{a}{,}\frac{{1}}{{2}}{}{b}{+}{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{{1}}{{2}}{}{b}{+}{a}{,}\frac{{1}}{{2}}{}{b}{-}{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{Γ}}{}\left({1}{-}{a}\right){}{\mathrm{HeunC}}{}\left({1}{,}{-}{b}{,}{1}{,}\frac{{1}}{{2}}{}{b}{-}{a}{+}\frac{{1}}{{2}}{,}{-}\frac{{1}}{{2}}{}{b}{+}{a}{,}{z}\right){}\left({1}{-}{z}\right)}{{\mathrm{Γ}}{}\left({1}{-}{b}\right)}{,}{\mathrm{cos}}{}\left({z}\right){=}\frac{{1}}{{2}}{}\frac{\left({2}{}{z}{+}{\mathrm{π}}\right){}{\mathrm{HeunC}}{}\left({1}{,}{1}{,}{1}{,}{0}{,}\frac{{1}}{{2}}{,}{-}{I}{}\left({2}{}{z}{+}{\mathrm{π}}\right)\right){}\left({I}{}\left({2}{}{z}{+}{\mathrm{π}}\right){+}{1}\right)}{{{ⅇ}}^{\frac{{1}}{{2}}{}{I}{}\left({2}{}{z}{+}{\mathrm{π}}\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)