Spheroidal Functions

SpheroidalS1[n,m,γ,z] and SpheroidalS2[n,m,γ,z]
radial spheroidal functions and
SpheroidalS1Prime[n,m,γ,z] and SpheroidalS2Prime[n,m,γ,z]
z derivatives of radial spheroidal functions
SpheroidalPS[n,m,γ,z] and SpheroidalQS[n,m,γ,z]
angular spheroidal functions and
SpheroidalPSPrime[n,m,γ,z] and SpheroidalQSPrime[n,m,γ,z]
z derivatives of angular spheroidal functions
SpheroidalEigenvalue[n,m,γ]spheroidal eigenvalue of degree n and order m

Spheroidal functions.

The radial spheroidal functions SpheroidalS1[n,m,γ,z] and SpheroidalS2[n,m,γ,z] and angular spheroidal functions SpheroidalPS[n,m,γ,z] and SpheroidalQS[n,m,γ,z] appear in solutions to the wave equation in spheroidal regions. Both types of functions are solutions to the equation . This equation has normalizable solutions only when is a spheroidal eigenvalue given by SpheroidalEigenvalue[n,m,γ]. The spheroidal functions also appear as eigenfunctions of finite analogs of Fourier transforms.

SpheroidalS1 and SpheroidalS2 are effectively spheroidal analogs of the spherical Bessel functions and , while SpheroidalPS and SpheroidalQS are effectively spheroidal analogs of the Legendre functions and . corresponds to a prolate spheroidal geometry, while corresponds to an oblate spheroidal geometry.

function
γ
z
range
name
angular prolate
radial prolate
angular oblate
radial oblate

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

Angular spheroidal functions can be viewed as deformations of Legendre functions.
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This plots angular spheroidal functions for various spheroidicity parameters.
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Angular spheroidal functions for integers are eigenfunctions of a band-limited Fourier transform.
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The Mathieu functions are a special case of spheroidal functions.

An angular spheroidal function with gives Mathieu angular functions.
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