Perturbation Solutions of the Whittaker-Hill Equation

When the Helmholtz equation ²V+k²V = 0 is separated in the generalparaboloidal co-ordinate system, the three ordinary differentialequations 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 solutionsof this equation. For k²>0, however, very little is so farknown. In this paper solutions of the Whittaker-Hill equationfor small positive k² are derived. These are the first explicitsolutions to be obtained for the case k²>0, and they couldbe employed to solve the Dirichlet or Neumann problem for ageneral paraboloid when k² is small. Three limiting cases arenoted, involving the reduction of the solutions to Mathieu functionsand the reduction of the co-ordinate system to the rotation-paraboloidalsystem.