Mathematica Tutorial

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 PS_{n, m} (, z) and QS_{n, m} (, z)
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 j_n (z) and y_n (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.

function			z	range	name
PS_{n, m} (, )	QS_{n, m} (, )			-1≤≤1	angular prolate
				≥1	radial prolate
PS_{n, m} (- , )	QS_{n, m} (- , )	-		-1≤≤1	angular oblate
		-		≥0	radial oblate