Abstract
Despite the great significance of equations with doubly-periodic coefficients in the methods of mathematical physics, the problem of solving Lame-Helmholtz equation still remains to be tackled. Arscott and Moglich method of double-series expansion as well as Malurkar nonlinear integral equation are incapable of reaching the final explicit solution. Our main result consists in obtaining analytic expressions for ellipsoidal wave functions of four species ℰci(sna), ℰsi (i=1,2,3,4) including the well known Lame functions Eci(sna), Esi(sna) as special cases. This is effected by deriving two integro-differential equations with variable coefficients and solving them by integral transform. Generalizing Riemann's idea of P function, we introduce D function to express their transformation properties.
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