Computation and Applications of Mathieu Functions: A Historical Perspective
Abstract
Mathieu functions of period $\pi$ or $2\pi$, also called elliptic cylinder functions, were introduced in 1868 by Émile 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 related to 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 antisecularity predates that of Lindstedt. For historical interest, we also provide short biographies of some of the major mathematical researchers involved in the history of the Mathieu functions: Émile Mathieu, Sir Edmund Whittaker, Edward Ince, and Gertrude Blanch.
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Submitted: 5 August 2020
Accepted: 7 July 2021
Published online: 4 November 2021
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Natural Sciences and Engineering Research Council of Canada https://doi.org/10.13039/501100000038 : RGPIN-2019-04749, RPIN-2020-06438
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/R014604/1
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