Spheroidal Functions

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

Spheroidal functions.

The radial spheroidal functions SpheroidalS1[n, m, , z] and SpheroidalS2[n, m, , z] and angular spheroidal functions SpheroidalPS[n, m, , z] and SpheroidalQS[n, m, , 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, ]. 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
z
range
name
angular prolate
radial prolate
angular oblate
radial oblate

Many different normalizations for spheroidal functions are used in the literature. Mathematica uses the Meixner-Schäfke 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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