convert to special functions of the Heun class - Maple 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

FunctionAdvisor( Heun );

The 23 functions in the "Heun" class are:

HeunB,HeunBPrime,HeunC,HeunCPrime,HeunD,HeunDPrime,HeunG,HeunGPrime,HeunT,HeunTPrime,MathieuA,MathieuB,MathieuC,MathieuCE,MathieuCEPrime,MathieuCPrime,MathieuExponent,MathieuFloquet,MathieuFloquetPrime,MathieuS,MathieuSE,MathieuSEPrime,MathieuSPrime

(1)
• 

convert/Heun accepts as optional arguments all those described in convert[to_special_function].

Examples

An assorted sample of special and elementary functions

functions_2F1:=ChebyshevT,JacobiP,SphericalY,EllipticK,GaussAGM,arctan,arcsin

functions_2F1:=ChebyshevT,JacobiP,SphericalY,EllipticK,GaussAGM,arctan,arcsin

(2)

Their syntax (calling sequence) in Maple

map2FunctionAdvisor,syntax,functions_2F1

ChebyshevTa,z,JacobiPa,b,c,z,SphericalYλ,μ,θ,φ,EllipticKk,GaussAGMx,y,arctany,x,arcsinz

(3)

A Heun representation for them, in these cases using HeunC

mapu→u=convertu,Heun,

ChebyshevTa,z=HeunC0,12,2a,0,a2+14,z1z+112+12za,JacobiPa,b,c,z=binomiala+b,bHeunC0,b,b+c+2a+1,0,12b+1+2ab+c+a+112b12ab+1,z1z+112+12zb+c+a+1,SphericalYλ,μ,θ,φ=121μ2λ+1πλμ!ⅇIφμcosθ+112μHeunC0,μ,2λ+1,0,λ2+λ+12,cosθ1cosθ+1λ+μ!cosθ112μΓ1μ12+12cosθλ+1,EllipticKk=12πHeunC0,0,0,0,14,k2k21k2+1,GaussAGMx,y=12x+y4xyx+y2HeunC0,0,0,0,14,14xy2yx,arctany,x=HeunC0,1,0,0,12,Iy+xx2+y2x2+y21+Iy+xx2+y2x2+y2y+Ix2+y2Ixx+Iy,arcsinz=zHeunC0,12,0,0,14,z2z21z2+1

(4)

A sample of special and elementary functions not admitting HeunG representation

functions_1F1:=erfz,dawsonz,Eia,z,LaguerreLa,b,z,hypergeoma,b,z,MeijerGa,,0,b,z,cosz,sinz

functions_1F1:=erfz,dawsonz,Eia,z,LaguerreLa,b,z,hypergeoma,b,z,MeijerGa,,0,b,z,cosz,sinz

(5)

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

mapu→u=convertu,Heun,functions_1F1

erfz=2zHeunB1,0,1,0,z2π,dawsonz=zHeunB1,0,1,0,z2ⅇz2,Eia,z=HeunB22a,0,2a,0,za1+za1Γ1a,LaguerreLa,b,z=binomiala+b,aHeunB2b,0,2+2b+4a,0,z,hypergeoma,b,z=HeunB2b2,0,2b4a,0,z,MeijerGa,,0,b,z=Γ1aHeunB2b,0,22b+4a,0,zΓ1b,cosz=122z+πHeunB2,0,0,0,I2z+πⅇ12I2z+π,sinz=zHeunB2,0,0,0,2IzⅇIz

(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

mapu→u=convertu,HeunC,functions_1F1

erfz=2z3+2zHeunC1,12,1,14,34,z2π,dawsonz=zHeunC1,12,1,14,34,z2z2+1ⅇz2,Eia,z=z+1HeunC1,1a,1,12a,12+12a,za1+za1Γ1a,LaguerreLa,b,z=binomiala+b,aHeunC1,b,1,1212ba,1+12b+a,zz+1,hypergeoma,b,z=HeunC1,b1,1,12b+a,12ba+12,zz+1,MeijerGa,,0,b,z=Γ1aHeunC1,b,1,12ba+12,12b+a,zz+1Γ1b,cosz=122z+πHeunC1,1,1,0,12,I2z+πI2z+π+1ⅇ12I2z+π,sinz=2Iz2+zHeunC1,1,1,0,12,2IzⅇIz

(7)

See Also

convert, convert[`1F1`], convert[`2F1`], convert[to_special_function], FunctionAdvisor, Heun functions, HeunB, HeunC, HeunG


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