Perturbation Solutions of the Whittaker-Hill Equation

When the Helmholtz equation ∇2V+k2V = 0 is separated in the general paraboloidal co-ordinate system, the three ordinary differential equations obtained each take, after a suitable change of variable, the form of the Whittaker-Hill equation. For the case k2<0, a considerable amount is known about the periodic solutions of this equation. For k2>0, however, very little is so far known. In this paper solutions of the Whittaker-Hill equation for small positive k2 are derived. These are the first explicit solutions to be obtained for the case k2>0, and they could be employed to solve the Dirichlet or Neumann problem for a general paraboloid when k2 is small. Three limiting cases are noted, involving the reduction of the solutions to Mathieu functions and the reduction of the co-ordinate system to the rotation-paraboloidal system.