Algebraic methods to compute Mathieu functions

Abstract

The standard form of the Mathieu differential equation is y''(z) + (a-2 qcos 2z) y(z) = 0, where a is the characteristic number and q is a real parameter. The most useful solution forms are given in terms of expansions for either small or large values of q. In this paper we obtain closed formulae for the generic term of expansions of Mathieu functions in the following cases: (1) standard series expansion for small q; (2) Fourier series expansion for small q; (3) asymptotic expansion in terms of trigonometric functions for large q; and (4) asymptotic expansion in terms of parabolic cylinder functions for large q. We also obtain closed formulae for the generic term of expansions of characteristic numbers and normalization formulae for small and large q. Using these formulae one can efficiently generate high-order expansions that can be used for implementation of the algebraic aspects of Mathieu functions in computer algebra systems. These formulae also provide alternative methods for numerical evaluation of Mathieu functions.