On Simplified Asymptotic Formulas for a Class of Mathieu Functions

Νaylor, D.

doi:10.1137/0515095

Cited by 7 publications

(1 citation statement)

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“…There are a large number of relatively modern books, papers, and software packages for the computation of the Mathieu functions. Analytical work on the Mathieu function is still extensive; to cite a paper almost at random, consider [55]. Some of the works ignore most of the others, and hence rediscover things, but others are very insightful.…”

Section: Selected Other Work By Other Contributorsmentioning

confidence: 99%

Computation and applications of Mathieu functions: A historical perspective

Brimacombe¹,

Corless²,

Zamir³

2020

Preprint

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Mathieu functions of period π or 2π, 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 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:

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Section: Selected Other Work By Other Contributorsmentioning

confidence: 99%