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convert/Heun - convert to special functions of the Heun class
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Calling Sequence
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convert(expr, Heun)
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Parameters
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expr
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Maple expression, equation, or a set or list of them
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Description
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convert/Heun converts, when possible, hypergeometric, MeijerG and special functions into Heun functions; that is, into one of
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FunctionAdvisor( Heun );
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The 23 functions in the "Heun" class are:
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![[HeunB, HeunBPrime, HeunC, HeunCPrime, HeunD, HeunDPrime, HeunG, HeunGPrime, HeunT, HeunTPrime, MathieuA, MathieuB, MathieuC, MathieuCE, MathieuCEPrime, MathieuCPrime, MathieuExponent, MathieuFloquet, MathieuFloquetPrime, MathieuS, MathieuSE, MathieuSEPrime, MathieuSPrime]](/support/helpjp/helpview.aspx?si=7084/file03609/math49.png)
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Examples
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An assorted sample of special and elementary functions
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![functions_2F1 := [ChebyshevT, JacobiP, SphericalY, EllipticK, GaussAGM, arctan, arcsin]](/support/helpjp/helpview.aspx?si=7084/file03609/math65.png)
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![functions_2F1 := [ChebyshevT, JacobiP, SphericalY, EllipticK, GaussAGM, arctan, arcsin]](/support/helpjp/helpview.aspx?si=7084/file03609/math68.png)
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Their syntax (calling sequence) in Maple
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![[ChebyshevT(a, z), JacobiP(a, b, c, z), SphericalY(lambda, mu, theta, phi), EllipticK(k), GaussAGM(x, y), arctan(y, x), arcsin(z)]](/support/helpjp/helpview.aspx?si=7084/file03609/math77.png)
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A Heun representation for them, in these cases using HeunC
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![map(proc (u) options operator, arrow; u = convert(u, Heun) end proc, [ChebyshevT(a, z), JacobiP(a, b, c, z), SphericalY(lambda, mu, theta, phi), EllipticK(k), GaussAGM(x, y), arctan(y, x), arcsin(z)])](/support/helpjp/helpview.aspx?si=7084/file03609/math89.png)
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A sample of special and elementary functions not admitting HeunG representation
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![functions_1F1 := [erf(z), dawson(z), Ei(a, z), LaguerreL(a, b, z), hypergeom([a], [b], z), MeijerG([[a], []], [[0], [b]], z), cos(z), sin(z)]](/support/helpjp/helpview.aspx?si=7084/file03609/math101.png)
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![functions_1F1 := [erf(z), dawson(z), Ei(a, z), LaguerreL(a, b, z), hypergeom([a], [b], z), MeijerG([[a], []], [[0], [b]], z), cos(z), sin(z)]](/support/helpjp/helpview.aspx?si=7084/file03609/math104.png)
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By default, the results are returned in terms of the lower Heun functions, that is, those with less parameters, in this case HeunB
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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
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