New method of solving Lame-Helmholtz equation and ellipsoidal wave functions

Despite the great significance of equations with doubly-periodic coefficients in the methods of mathematical physics, the problem of solving Lamé-Helmholtz equation still remains to be tackled. Arscott and Möglich 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 _ci(sna), _si (i=1,2,3,4) including the well known Lamé functions E_ci(sna), E_si(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.