Mathieu函数的计算与应用：历史的展望
Computation and applications of Mathieu functions: A historical perspective
摘要：
周期为$π$或$2π$的马蒂厄函数，也称为椭圆柱函数，是由埃米尔·马蒂厄在1868年与所谓的修正马蒂厄函数一起引入的，目的是帮助理解固定椭圆环中弹性膜的振动。这些功能仍然经常出现在今天的应用程序：我们的兴趣，例如，刺激了一个问题的脉动血流在血管压缩成一个椭圆形的横截面。本文综述和概括了Mathieu函数和修正Mathieu函数的理论和计算方法的历史发展，并指出了当前软件能力中的一些差距，特别是与Mathieu方程的双特征值有关的问题。我们演示了如何计算Puienix展开的Mathieu特征值关于这样的双重特征值，并给出方法来计算广义本征函数出现在那里。在审查马修的原始贡献，我们带出，他使用的反世俗早于林德施泰特。出于兴趣，我们还提供了简短的传记，一些主要的数学研究人员参与历史上的马蒂厄职能：埃米尔马蒂厄，爵士埃德蒙惠特克，爱德华因斯，和格特鲁德布兰奇。
英文摘要：
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: Émile Mathieu, Sir Edmund Whittaker, Edward Ince, and Gertrude Blanch.
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