Computation and applications of Mathieu functions: A historical perspective

TL;DR

This paper reviews the historical development and computational methods for Mathieu functions, highlighting current software gaps and presenting new techniques for handling double eigenvalues and generalized eigenfunctions.

Contribution

It provides a comprehensive historical overview and introduces novel computational methods for Mathieu functions, especially for double eigenvalues, filling gaps in existing software capabilities.

Findings

Identification of gaps in current Mathieu function software

Development of methods for Puiseux expansions of eigenvalues

Techniques for computing generalized eigenfunctions at double eigenvalues

Abstract

Mathieu functions of period $\pi$ or $2\pi$, also called elliptic cylinder functions, were introduced in 1868 by \'Emile Mathieu together with so-called modified Mathieu functions, in order to help understand the vibrations of an elastic membrane set in a fixed elliptical hoop. These functions still occur frequently in applications today: our interest, for instance, was stimulated by a problem of pulsatile blood flow in a blood vessel compressed into an elliptical cross-section. This paper surveys and recapitulates the historical development of the theory and methods of computation for Mathieu functions and modified Mathieu functions and identifies some gaps in current software capability, particularly to do with double eigenvalues of the Mathieu equation. We demonstrate how to compute Puiseux expansions of the Mathieu eigenvalues about such double eigenvalues, and give methods to compute the generalized eigenfunctions that arise there. In examining Mathieu's original contribution, we bring out that his use of anti-secularity predates that of Lindstedt. For interest, we also provide short biographies of some of the major mathematical researchers involved in the history of the Mathieu functions: \'Emile Mathieu, Sir Edmund Whittaker, Edward Ince, and Gertrude Blanch.