info prev up next book cdrom email home

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

\begin{displaymath}
c\equiv {\textstyle{1\over 2}}ak.
\end{displaymath} (2)

Substitute in a trial solution
\begin{displaymath}
\Phi=R_{mn}(c,\xi_1)S_{mn}(c,\xi_2){\cos \atop \sin}(m\phi )
\end{displaymath} (3)


\begin{displaymath}
{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.
\end{displaymath} (4)

The radial differential equation is


\begin{displaymath}
{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,
\end{displaymath} (5)

and the angular differential equation is


\begin{displaymath}
{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.
\end{displaymath} (6)

Note that these are identical (except for a sign change). The prolate angular function of the first kind is given by
\begin{displaymath}
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}
\end{displaymath} (7)

where $P_m^k(\eta)$ is an associated Legendre Polynomial. The prolate angular function of the second kind is given by
\begin{displaymath}
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}
\end{displaymath} (8)

where $Q^m_k(\eta)$ is an associated Legendre Function of the Second Kind and the Coefficients $d_r$ satisfy the Recurrence Relation
\begin{displaymath}
\alpha_kd_{k+2}+(\beta_k-\lambda_{mn})d_k+\gamma_kd_{k-2}=0,
\end{displaymath} (9)

with


$\displaystyle \alpha_k$ $\textstyle =$ $\displaystyle {(2m+k+2)(2m+k+1)c^2\over (2m+2k+3)(2m+2k+5)}$ (10)
$\displaystyle \beta_k$ $\textstyle =$ $\displaystyle (m+k)(m+k+1)+{2(m+k)(m+k+1)-2m^2-1\over (2m+2k-1)(2m+2k+3)}c^2$ (11)
$\displaystyle \gamma_k$ $\textstyle =$ $\displaystyle {k(k-1)c^2\over (2m+2k-3)(2m+2k-1)}.$ (12)


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

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

Stratton et al. (1956) use
\begin{displaymath}
{(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}
\end{displaymath} (14)

Flammer (1957) uses
\begin{displaymath}
S_{mn}(c,0)=\cases{
P_n^{m+1}(0) & for $n-m$\ odd\cr
P_n^m(0) & for $n-m$\ even.\cr}
\end{displaymath} (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.



info prev up next book cdrom email home

© 1996-9 Eric W. Weisstein
1999-05-26