Abstract
The Whittaker-Hill equation arises when Helmholtz's equation v[2]v + k[2]v = 0 is separated in general paraboloidal coordinates. Paraboloidal wave functions are (certain) solutions of the Whittaker-Hill equation, with period pi or pi. Chapter I is introductory: the general paraboloidal coordinate system and the separation of Helmholtz's equation are discussed. Chapters II and III deal with the case k[2]<0. Chapter II gives the basic properties of the paraboloidal wave functions, together with some original results on orthogonality and simple integral properties. Chapter III is devoted to double integral equations for the paraboloidal wave functions. Chapters IV and V deal with the case k[2]>0: the theory for k[2]<0 does not carry over immediately, so that a different treatment is needed. Chapter IV discusses the form of the periodic solutions, the continuity of the characteristic values, orthogonality properties, simple relations between the solutions and some degenerate forms of the differential equation and its solutions. In Chapter V explicit perturbation solutions are given, together with perturbation series for the characteristic values. The convergence of these latter series is discussed, and they are used to show the relation, for small values of the parameters involved, between the solutions of the previous chapter.