New Method of Solving Lame-Helmholtz Equation and Ellipsoidal Wave Functions

Dong Ming-de

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Dong Ming-de. New Method of Solving Lame-Helmholtz Equation and Ellipsoidal Wave Functions[J]. Applied Mathematics and Mechanics, 1984, 5(2): 167-178.

Citation:

Dong Ming-de. New Method of Solving Lame-Helmholtz Equation and Ellipsoidal Wave Functions[J]. Applied Mathematics and Mechanics, 1984, 5(2): 167-178.

Citation:

Dong Ming-de. New Method of Solving Lame-Helmholtz Equation and Ellipsoidal Wave Functions[J]. Applied Mathematics and Mechanics, 1984, 5(2): 167-178.

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New Method of Solving Lame-Helmholtz Equation and Ellipsoidal Wave Functions

Dong Ming-de

Institute of Theoretical Physics, Academia Sinica, Beijing

Received Date: 1982-12-10
Publish Date: 1984-04-15

Abstract

Abstract

Despite the great significance of equations with doubly-periodic coefficients in the methods of mathematical physics, the problem of solving Lamb-Helmholtz equation still remains to be tackled; Arscott and Möeglich's method of double-series expansion as well as Malurkar's non-linear integral equation are unable to reach the final solution, Our main result consists in obtaining analytic expression for ellipsoidal wave funcdons of four species ε_ci(sna),ε_si(sna)(i=1,2,3,4) by deriving a couple of linear integral equations and solving these by integral transform, including the well-known Lamé function E_ci(sna),E_si(sna) as special case.Generalizing Riemann's idea of P-function, we introduce D-function to eapress their transformation properties.

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References(7)

References

[1]	Erdelyi,Higher Transcendental Functions (Bateman Manuscript Project).Vol.Ⅰ-Ⅲ (1953-1955).
[2]	Whittaker, Watson, Modern Analgsis,Cambridge University Press(1940).
[3]	Hobson, E,W.,Theory of Spherical and Ellipsosdal Harmonics,Cambridge University Press, (1931).
[4]	Möglich,Beugunsercheinungen an Körpern von Ellipsoidischer Gestalt,Ann, d,Phgs,83(1927),609-734.
[5]	Arscott,F M.,(a) Periodic Differential Equations,Pergamon Press(1964).(b) A new treatment of the ellipsoidal wave equations, Proc, Lond.Math Soc.,33 (1959),21-50.
[6]	Malurkar,Ellipsoidal wave functions, Ind, J Phqs,9(1935),45-80.
[7]	Dong Ming-de,Poincarés Problem of Irregular Integrals (Lecture Notes, unpublished (1981)).