## On the numerical construction of ellipsoidal wave functions

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- by F. M. Arscott, P. J. Taylor and R. V. M. Zahar PDF
- Math. Comp.
**40**(1983), 367-380 Request permission

## Abstract:

The ellipsoidal wave equation, which is the most general equation derived by separation of the Helmholtz equation in confocal coordinates, presents unusual computational difficulties, and its solutions, despite their importance for physical applications, have not hitherto been effectively computed. This paper describes a successful technique, which involves the solution of a four-term recursion and the simultaneous handling of two eigenparameters.## References

- F. M. Arscott,
*Perturbation solutions of the ellipsoidal wave equation*, Quart. J. Math. Oxford Ser. (2)**7**(1956), 161–174. MR**94484**, DOI 10.1093/qmath/7.1.161 - F. M. Arscott,
*A new treatment of the ellipsoidal wave equation*, Proc. London Math. Soc. (3)**9**(1959), 21–50. MR**104837**, DOI 10.1112/plms/s3-9.1.21
F. M. Arscott, - F. M. Arscott,
*Neumann-series solutions of the ellipsoidal wave equation*, Proc. Roy. Soc. Edinburgh Sect. A**67**(1963/65), 69–77. MR**187579** - F. M. Arscott and I. M. Khabaza,
*Tables of Lamé polynomials*, A Pergamon Press Book, The Macmillan Company, New York, 1962. MR**0152680** - F. M. Arscott, R. Lacroix, and W. T. Shymanski,
*A three-term recursion and the computation of Mathieu functions*, Proceedings of the Eighth Manitoba Conference on Numerical Mathematics and Computing (Univ. Manitoba, Winnipeg, Man., 1978) Congress. Numer., XXII, Utilitas Math., Winnipeg, Man., 1979, pp. 107–115. MR**541916** - F. M. Arscott and B. D. Sleeman,
*High-frequency approximations to ellipsoidal wave functions*, Mathematika**17**(1970), 39–46. MR**270003**, DOI 10.1112/S0025579300002680 - Robert Campbell,
*Sur la vibration d’un haut-parleur elliptique*, C. R. Acad. Sci. Paris**228**(1949), 970–972 (French). MR**29454** - W. Gautschi,
*Zur Numerik rekurrenter Relationen*, Computing (Arch. Elektron. Rechnen)**9**(1972), 107–126 (German, with English summary). MR**312714**, DOI 10.1007/bf02236961 - B. A. Hargrave and B. D. Sleeman,
*Uniform asymptotic expansions for ellipsoidal wave functions. I. High frequency solutions of the ellipsoidal wave equations*, J. Inst. Math. Appl.**14**(1974), 31–40. MR**419891** - G. W. Hill,
*On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon*, Acta Math.**8**(1886), no. 1, 1–36. MR**1554690**, DOI 10.1007/BF02417081 - E. L. Ince,
*Ordinary Differential Equations*, Dover Publications, New York, 1944. MR**0010757**
E. L. Ince, "Tables of the elliptic-cylinder functions," - R. V. M. Zahar,
*Recurrence techniques for a differential eigenvalue problem*, Proceedings of the Eighth Manitoba Conference on Numerical Mathematics and Computing (Univ. Manitoba, Winnipeg, Man., 1978) Congress. Numer., XXII, Utilitas Math., Winnipeg, Man., 1979, pp. 479–485. MR**541940**

*Periodic Differential Equations*, Pergamon Press, New York, 1964.

*Proc. Roy. Soc. Edinburgh*, v. 52, 1932, pp. 335-423. E. L. Ince, "Zeros and turning-points of the elliptic-cylinder functions,"

*Proc. Roy. Soc. Edinburgh*, v. 52, 1932, pp. 424-433. E. Mathieu, "Mémoire sur le mouvement vibratoire d’une membrane de forme elliptique,"

*J. Math. Pures Appl.*, v. 13, 1868, pp. 137-203. R. V. M. Zahar, "A mathematical analysis of Miller’s algorithm,"

*Numer. Math.*, v. 27, 1977, pp. 427-447.

## Additional Information

- © Copyright 1983 American Mathematical Society
- Journal: Math. Comp.
**40**(1983), 367-380 - MSC: Primary 65D20; Secondary 33A60, 65H10, 65P05
- DOI: https://doi.org/10.1090/S0025-5718-1983-0679452-6
- MathSciNet review: 679452