SIAM Review, Volume 63, Issue 4, Page 653-720, January 2021.
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.

中文翻译：

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Mathieu 函数的计算和应用：历史视角

SIAM 评论，第 63 卷，第 4 期，第 653-720 页，2021 年 1 月。
周期 $\pi$ 或 $2\pi$ 的 Mathieu 函数，也称为椭圆柱面函数，由 Émile Mathieu 于 1868 年与所谓的修正 Mathieu 函数一起引入，以帮助理解弹性膜的振动。固定椭圆箍。这些功能在今天的应用中仍然频繁出现；例如，我们的兴趣是由一个被压缩成椭圆形横截面的血管中的脉动血流问题激发的。本文调查并概括了 Mathieu 函数和修正 Mathieu 函数的理论和计算方法的历史发展，并确定了当前软件能力中的一些差距，特别是与 Mathieu 方程的双特征值相关的差距。我们演示了如何计算关于这种双特征值的 Mathieu 特征值的 Puiseux 展开，并给出计算在那里出现的广义特征函数的方法。在检查马修的原始贡献时，我们发现他对反世俗的使用早于林斯泰特的使用。出于历史兴趣，我们还提供了一些参与 Mathieu 函数历史的主要数学研究人员的简短传记：Émile Mathieu、Edmund Whittaker、Edward Ince 和 Gertrude Blanch。