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Prolate Spheroidal Wave Function

The wave equation in prolate spheroidal coordinates is

del ^2Phi+k^2Phi=partial/(partialxi_1)[(xi_1^2-1)(partialPhi)/(partialxi_1)]+partial/(partialxi_2)[(1-xi_2^2)(partialPhi)/(partialxi_2)] +(xi_1^2-xi_2^2)/((xi_1^2-1)(1-xi_2^2))(partial^2Phi)/(partialphi^2)+c^2(xi_1^2-xi_2^2)Phi=0,
(1)

where

c=1/2ak.
(2)

Substitute in a trial solution

Phi=R_(mn)(c,xi_1)S_(mn)(c,xi_2)cos; sin(mphi)
(3)
d/(dxi_1)[(xi_1^2-1)d/(dxi_1)R_(mn)(c,xi_1)]-(lambda_(mn)-c^2xi_1^2+(m^2)/(xi_1^2-1))R_(mn)(c,xi_1)=0.
(4)

The radial differential equation is

d/(dxi_2)[(xi_2^2-1)d/(dxi_2)S_(mn)(c,xi_2)]-(lambda_(mn)-c^2xi_2^2+(m^2)/(xi_2^2-1))R_(mn)(c,xi_2)=0,
(5)

and the angular differential equation is

d/(dxi_2)[(1-xi_2^2)d/(dxi_2)S_(mn)(c,xi_2)]-(lambda_(mn)-c^2xi_2^2+(m^2)/(1-xi_2^2))S_(mn)(c,xi_2)=0.
(6)

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

S_(mn)^((1))={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,
(7)

where P_m^k(eta) is an associated Legendre polynomial. The prolate angular function of the second kind is given by

S_(mn)^((2))={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,
(8)

where Q_k^m(eta) is an associated Legendre function of the second kind and the coefficients d_r satisfy the recurrence relation

alpha_kd_(k+2)+(beta_k-lambda_(mn))d_k+gamma_kd_(k-2)=0,
(9)

with

alpha_k=((2m+k+2)(2m+k+1)c^2)/((2m+2k+3)(2m+2k+5))
(10)
beta_k=(m+k)(m+k+1)+(2(m+k)(m+k+1)-2m^2-1)/((2m+2k-1)(2m+2k+3))c^2
(11)
gamma_k=(k(k-1)c^2)/((2m+2k-3)(2m+2k-1)).
(12)

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

int_(-1)^1[S_(mn)(c,eta)]^2deta=2/(2n+1)((n+m)!)/((n-m)!).
(13)

Stratton et al. (1956) use

((n+m)!)/((n-m)!)={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.
(14)

Flammer (1957) uses

S_(mn)(c,0)={P_n^(m+1)(0) for n-m odd; P_n^m(0) for n-m even.
(15)

See also

Oblate Spheroidal Wave Function, Spheroidal Wave Function

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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.

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Prolate Spheroidal Wave Function

Cite this as:

Weisstein, Eric W. "Prolate Spheroidal Wave Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ProlateSpheroidalWaveFunction.html

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