The wave equation in prolate spheroidal coordinates is

(1)

where

(2)

Substitute in a trial solution

(3)

(4)

The radial differential equation is

(5)

and the angular differential equation is

(6)

Note that these are identical (except for a sign change). The prolate angular function of the first kind is given by

{sum_(r=1,3,...)^inftyd_r(c)P_(m+r)^m(eta) for n-m odd; sum_(r=0,2,...)^inftyd_r(c)P_(m+r)^m(eta) for n-m even, " src="http://mathworld.wolfram.com/images/equations/ProlateSpheroidalWaveFunction/NumberedEquation7.gif" style="height:64px; width:285px" />

(7)

where  is an associated Legendre polynomial. The prolate angular function of the second kind is given by

{sum_(r=...,-1,1,3,...)d_r(c)Q_(m+r)^m(eta) for n-m odd; sum_(r=...,-2,0,2,...)d_r(c)Q_(m+r)^m(eta) for n-m even, " src="http://mathworld.wolfram.com/images/equations/ProlateSpheroidalWaveFunction/NumberedEquation8.gif" style="height:86px; width:293px" />

(8)

where  is an associated Legendre function of the second kind and the coefficients  satisfy the recurrence relation

(9)

with

			(10)
			(11)
			(12)

Various normalization schemes are used for the s (Abramowitz and Stegun 1972, p. 758). Meixner and Schäfke (1954) use

(13)

Stratton et al. (1956) use

{sum_(r=1,3,...)^(infty)((r+2m)!)/(r!)d_r for n-m odd; sum_(r=0,2,...)^(infty)((r+2m)!)/(r!)d_r for n-m even. " src="http://mathworld.wolfram.com/images/equations/ProlateSpheroidalWaveFunction/NumberedEquation11.gif" style="height:102px; width:293px" />

(14)

Flammer (1957) uses

{P_n^(m+1)(0) for n-m odd; P_n^m(0) for n-m even. " src="http://mathworld.wolfram.com/images/equations/ProlateSpheroidalWaveFunction/NumberedEquation12.gif" style="height:46px; width:230px" />

(15)

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Spheroidal Wave Functions." Ch. 21 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 751-759, 1972.

Flammer, C. Spheroidal Wave Functions. Stanford, CA: Stanford University Press, 1957.

Meixner, J. and Schäfke, F. W. Mathieusche Funktionen und Sphäroidfunktionen. Berlin: Springer-Verlag, 1954.

Rhodes, D. R. "On the Spheroidal Functions." J. Res. Nat. Bur. Standards--B. Math. Sci. 74B, 187-209, Jul.-Sep. 1970.

Stratton, J. A.; Morse, P. M.; Chu, L. J.; Little, J. D. C.; and Corbató, F. J. Spheroidal Wave Functions. New York: Wiley, 1956.