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
jn (z) and
yn (z), while
SpheroidalPS and
SpheroidalQS are effectively spheroidal analogs of the Legendre functions

and

.
2>0 corresponds to a
prolate spheroidal geometry, while
2<0 corresponds to an
oblate spheroidal geometry.
Many different normalizations for spheroidal functions are used in the literature.
Mathematica uses the Meixner-Schäfke normalization scheme.
The Mathieu functions are a special case of spheroidal functions.