论文标题
Mathieu功能的计算和应用：历史观点
Computation and applications of Mathieu functions: A historical perspective
论文作者
论文摘要
1868年，émileMathieu以及所谓的修改Mathieu功能引入了$π$或$2π$的Mathieu功能，也称为椭圆缸功能，以帮助了解固定椭圆形箍中设置的弹性膜的振动。当今的应用中，这些功能仍然经常出现：例如，我们的兴趣受到压缩到椭圆形横截面的血管中脉冲血流的问题刺激。本文调查并概括了Mathieu功能的理论和计算方法的历史发展，并修改了Mathieu函数，并确定了当前软件能力的某些差距，尤其是与Mathieu方程的双重特征值有关。我们演示了如何计算Mathieu特征值的Puiseux扩展，并提供了计算那里出现的广义特征功能的方法。在研究Mathieu的最初贡献时，我们表明他对反临界的使用早于Lindstedt的使用。为了兴趣，我们还提供了一些参与Mathieu功能历史的主要数学研究人员的简短传记：émileMathieu，Edmund Whittaker爵士，Edward Ince和Gertrude Blanch。
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.
