III.—Theory of the Whittaker Hill Equation

When the Helmholtz equation ∇ 2 V + k 2 V = o is separated in the general paraboloidal coordinate 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 k 2 < o, a considerable amount is known about the periodic solutions of this equation. The theory for k 2 < o does not carry over immediately to the case k 2 < o, and so far only perturbation solutions have been obtained. This paper gives, in sections 1–5, explicit solutions for the case k 2 < o, in the form of trigonometric series determined by three term recurrence relations. In sections 6, 7 some important relations and orthogonality properties are discussed, which are of particular significance in respect of application to boundary value problems. Section 8 discusses some degenerate cases, and in the Appendix an important property is established of continuity of the solutions with respect to the parameters of the equation.