Prolate Spheroidal Wave Function

The Wave Equation in Prolate Spheroidal Coordinates is

$\nabla^2\Phi+k^2\Phi={\partial \over \partial \xi_1} \left[{({\xi_1}^2-1){\part... ...\partial \xi_2} \left[{(1-{\xi_2}^2){\partial \Phi\over \partial \xi_2}}\right]$
$\mathop{+}{{\xi_1}^2-{\xi_2}^2\over ({\xi_1}^2-1)(1-{x_2}^2)} {\partial ^2\Phi\over \partial \phi ^2} + c^2({\xi_1}^2-{\xi_2}^2)\Phi=0,\quad$	(1)

where

$$c\equiv {\textstyle{1\over 2}}ak.$$ (2)

Substitute in a trial solution

$$\Phi=R_{mn}(c,\xi_1)S_{mn}(c,\xi_2){\cos \atop \sin}(m\phi )$$ (3)

$${d\over d\xi_1} \left[{({\xi_1}^2-1){d\over d\xi_1} R_{mn}(c... ...-c^2{\xi_1}^2+{m^2\over {\xi_1}^2-1}}\right)R_{mn}(c,\xi_1)=0.$$ (4)

The radial differential equation is

$${d\over d\xi_2} \left[{({\xi_2}^2-1){d\over d\xi_2} S_{mn}(c... ...-c^2{\xi_2}^2+{m^2\over {\xi_2}^2-1}}\right)R_{mn}(c,\xi_2)=0,$$ (5)

and the angular differential equation is

$${d\over d\xi_2} \left[{(1-{\xi_2}^2){d\over d\xi_2} S_{mn}(c... ...-c^2{\xi_2}^2+{m^2\over 1-{\xi_2}^2}}\right)R_{mn}(c,\xi_2)=0.$$ (6)

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

$$S_{mn}^{(1)}=\cases{ \sum\nolimits_{r=1, 3, \ldots}^\infty ... ...2, \ldots}^\infty d_r(c)P^m_{m+r}(\eta) & for $n-m$\ even,\cr}$$ (7)

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

$$S_{mn}^{(2)}=\cases{ \sum\limits_{r=\ldots, -1, 1, 3, \ldot... ...-2, 0, 2, \ldots} d_r(c)Q^m_{m+r}(\eta) & for $n-m$\ even,\cr}$$ (8)

where $Q^m_k(\eta)$ 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

$$\alpha_k = {(2m+k+2)(2m+k+1)c^2\over (2m+2k+3)(2m+2k+5)}$$ (10)

$$\beta_k = (m+k)(m+k+1)+{2(m+k)(m+k+1)-2m^2-1\over (2m+2k-1)(2m+2k+3)}c^2$$ (11)

$$\gamma_k = {k(k-1)c^2\over (2m+2k-3)(2m+2k-1)}.$$ (12)

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

$$\int_{-1}^1 [S_{mn}(c,\eta)]^2\,d\eta = {2\over 2n+1} {(n+m)!\over (n-m)!}.$$ (13)

Stratton et al. (1956) use

$${(n+m)!\over (n-m)!} = \cases{ \sum_{r=1, 3, \ldots}^\infty... ...2, \ldots}^\infty {(r+2m)!\over r!} d_r & for $n-m$\ even.\cr}$$ (14)

Flammer (1957) uses

$$S_{mn}(c,0)=\cases{ P_n^{m+1}(0) & for $n-m$\ odd\cr P_n^m(0) & for $n-m$\ even.\cr}$$ (15)

See also Oblate Spheroidal Wave Function

References

Abramowitz, M. and Stegun, C. 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.

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