The Tabulation of Mathieu Functions

and in which a and 6 are constants, arises in applied mathematics in two main groups of problems. The first group, of which Mathieu's problem of the transverse vibrations of a taut elliptic membrane is typical, consists of boundary problems relative to some partial differential equation, such as the wave equation, where the boundary is an ellipse—or possibly a hyperbola. Here a fundamental requirement is usually that y shall admit the period 2w in x. With a prescribed value of 6, this restricts a to a discrete set of characteristic numbers, each associated, in the concrete problem, with the frequency of a normal mode of vibration or some analogous quantity. In the complete solution of this class of problem, solutions of the associated equation