Spheroidal Wavefunction

Whittaker and Watson (1990, p. 403) define the internal and external spheroidal wavefunctions as

$$S_{mn}^{(1)} = 2\pi {(n-m)!\over (n+m)!} P_n^m(ir)P_n^m(\cos\theta)\begin{array}{c}\cos\\ \sin\end{array}(m\phi)$$

$$S_{mn}^{(2)} = 2\pi {(n-m)!\over (n+m)!} Q_n^m(ir)Q_n^m(\cos\theta)\begin{array}{c}\cos\\ \sin\end{array}(m\phi).$$

See also Ellipsoidal Harmonic, Oblate Spheroidal Wave Function, Prolate Spheroidal Wave Function, Spherical Harmonic

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.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 642-644, 1953.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.