Mathieu Functions
special functions introduced by the French mathematician E. Mathieu in 1868 in the course of his work on the vibrations of an elliptical membrane. Mathieu functions are also used in the study of the propagation of electromagnetic waves in an elliptical cylinder and in the examination of tidal waves in a container having the shape of an elliptical cylinder. Even or odd functions that are periodic solutions of the linear second-order differential equation (Mathieu’s equation)
are termed Mathieu functions. The condition that the solution of this equation be periodic determines a sequence of values for λ dependent on q. If q = 0, then λ = n2 (n = 1, 2, …), and the Mathieu functions in this case are cos nz and sin nz. For q ≠ 0, the Mathieu functions, denoted by cen(z, q) and sen(z, q), can be represented in the form
where ank and bnk depend on q; ε = 0 for even n and ε = 1 for odd n.
REFERENCES
Whittaker, E. T., and G. N. Watson.Kurs sovremennogo analiza, part 2, 2nd ed. Moscow, 1963. (Translated from English.)McLachlan, N. W. Teoriia i prilozhentiia funktsii Mat’e. Moscow, 1953. (Translated from English.)
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