Spheroidal Functions

Spheroidal functions.

The radial spheroidal functions (RefLink[SpheroidalS1, paclet:ref/SpheroidalS1])[n,m,γ,z] and (RefLink[SpheroidalS2, paclet:ref/SpheroidalS2])[n,m,γ,z] and angular spheroidal functions (RefLink[SpheroidalPS, paclet:ref/SpheroidalPS])[n,m,γ,z] and (RefLink[SpheroidalQS, paclet:ref/SpheroidalQS])[n,m,γ,z] appear in solutions to the wave equation in spheroidal regions. Both types of functions are solutions to the equation (1-(z^2))y''-2zy'+(λ+(γ^2)(1-(z^2))-((m^2)/(1-(z^2))))y==0. This equation has normalizable solutions only when λ is a spheroidal eigenvalue given by (RefLink[SpheroidalEigenvalue, paclet:ref/SpheroidalEigenvalue])[n,m,γ]. The spheroidal functions also appear as eigenfunctions of finite analogs of Fourier transforms.

(RefLink[SpheroidalS1, paclet:ref/SpheroidalS1]) and (RefLink[SpheroidalS2, paclet:ref/SpheroidalS2]) are effectively spheroidal analogs of the spherical Bessel functions (Subscript[j, n])(z) and (Subscript[y, n])(z), while (RefLink[SpheroidalPS, paclet:ref/SpheroidalPS]) and (RefLink[SpheroidalQS, paclet:ref/SpheroidalQS]) are effectively spheroidal analogs of the Legendre functions (Subscript[P, n]^m)(z) and (Subscript[Q, n]^m)(z). (γ^2)>0 corresponds to a prolate spheroidal geometry, while (γ^2)<0 corresponds to an oblate spheroidal geometry.

({{function, , γ, z, range, name}, {P(Subscript[S, n,m])(γ,η), Q(Subscript[S, n,m])(γ,η), γ, η, -1≤η≤1, angular prolate}, {(Subscript[S, n,m]^((1)))(γ,ζ), (Subscript[S, n,m]^((2)))(γ,ζ), γ, ζ, ζ≥1, radial prolate}, {P(Subscript[S, n,m])(-i γ,η), Q(Subscript[S, n,m])(-i γ,η), -i γ, η, -1≤η≤1, angular oblate}, {(Subscript[S, n,m]^((1)))(-i γ,i ζ), (Subscript[S, n,m]^((2)))(-i γ,i ζ), -i γ, i ζ, ζ≥0, radial oblate}})

Many different normalizations for spheroidal functions are used in the literature. The Wolfram System uses the Meixner–Schäfke normalization scheme.

Angular spheroidal functions can be viewed as deformations of Legendre functions:

Series[SpheroidalPS[n,0,γ,η],{γ,0,3}]

This plots angular spheroidal functions for various spheroidicity parameters:

Plot[{SpheroidalPS[3,0,0,η],SpheroidalPS[3,0,3,η],SpheroidalPS[3,0,5,η]},{η,-1,1}]

Angular spheroidal functions (Subscript[PS, n,0])(γ,η) for integers n≥0 are eigenfunctions of a band-limited Fourier transform:

Integrate[SpheroidalPS[3,0,γ,η]Exp[I ω γ η],{η,-1,1}]

The Mathieu functions are a special case of spheroidal functions.

An angular spheroidal function with m==(1/2) gives Mathieu angular functions:

SpheroidalPS[1/2,1/2,c,z]