# CARMA Colloquium

## 4:00 pm

## Thursday, 25^{th} Mar 2021

**SR118, SR Building (and online via Zoom)**

Join via Zoom, or join us in person (max room capacity is 9 people).

3:30pm for pre-talk drinks + snacks, and 4pm for the talk

You can watch a video version at https://youtu.be/0rE-EopdSyQ instead, or in addition!

# Prof Robert Corless

(University of Western Ontario)
*Computation and Application of Mathieu Functions: a Survey from a Historical Point of View*

A full paper describing this talk can be found at

https://arxiv.org/abs/2008.01812. 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 talk surveys and recapitulates some of 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.