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
and
SpheroidalS2
and
angular spheroidal functions SpheroidalPS
and
SpheroidalQS
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
. 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.
| | | | |
 |  |  |  |  | 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.
| Out[1]= |  |
This plots angular spheroidal functions for various spheroidicity parameters.
| Out[2]= |  |
Angular spheroidal functions
for integers
are eigenfunctions of a band-limited Fourier transform.
| Out[3]= |  |
The Mathieu functions are a special case of spheroidal functions.
An angular spheroidal function with
gives Mathieu angular functions.
| Out[4]= |  |
