  Mathieu functions of period $\pi$ or $2\pi$, also called elliptic cylinder
functions, were introduced in 1868 by \'Emile 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: \'Emile Mathieu,
Sir Edmund Whittaker, Edward Ince, and Gertrude Blanch.
