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Mathieu functions are obviously solutions to the Mathieu differential
equation and were investigated by Emile Leonard Mathieu (1835–1890), who was studying free oscillations of an elliptic membrane. These functions were further investigated by others, and a collection of results was published over 60 years ago by McLachlan, which is still a standard reference. The crucial feature that sets Mathieu functions apart from other second-order differential equations is that they have periodic coefficients.
These functions arise in two main areas: applications involving elliptical geometries and problems involving periodic motion. Examples of the first area include analysis of vibrational modes in elliptical membranes, propagating modes in elliptic waveguides, etc. The second area includes the trajectory of an electron in a periodic array of atoms, oscillations of floating boats, charged particle traps, etc.
In terms of pedagogy, with very few exceptions, most books on mathematical methods or special functions do not deal with Mathieu functions, possibly because of the complexity of the solutions. Notable exceptions include the books by Morse and Feshbach and Mathews and Walker. The reader is warned that a variety of notations are used in the literature (see Sec. 11.10). Mathieu functions are also known as elliptical cylinder functions. In this chapter, we will present a survey of Mathieu functions.
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