关于Lamé-Helmholtz方程的新解法和椭球波动函数

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

摘要: 双周期系数方程虽在数理方法中具有重要意义,但Lamé—Helmholtz方程的解至今仍来求出,因为Arscott和Möglich的双重级数展开法,Malurkar的非线性积分方程都无法进一步处理本文的主要结果是由原方程导出一组线性微积分方程,利用积分变换,直接求得四类椭球波动函数,ε_ci(sna),ε_si(sna)(i=1,2,3,4),它的特例就是熟知的Lamé函数E_ci(sna),E_si(sna),推广Riemann P函数思想,引进D函数来表示其变换规律。

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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