# Mathieu Functions

## Mathieu functions

[ma′tyü ‚fəŋk·shənz]
(mathematics)
Any solution of the Mathieu equation which is periodic and an even or odd function.

## 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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The elements of the layer matrix which characterizes the reflection grating are calculated in terms of Mathieu functions.
Portugal, "Algebraic methods to compute Mathieu functions," J.
The solution of such an equation invariably leads to Mathieu functions [Alhargan 1996; Alhargan and Judah 1992; McLachlan 1947; Morse and Feshbach 1953; NBS 1967].
Many algorithms have been devised to compute Mathieu functions with various degrees of success [Alhargan 1996; Arscott and Shymansky 1978; Blanch 1966; Leeb 1979; McLachlan 1947; Morse and Feshbach 1953; NBS 1967; Rengarajan and Lewis 1980; Shirts 1993a; 1993b; Toyama and Shogan 1984; Wimp 1984].
The objective of the article is to present all the necessary algorithms for computing Mathieu functions without needing to recourse to the literature.
Equation (3) has two solutions, termed radial first-kind and radial second-kind Mathieu functions: the radial first-kind even Mathieu function
4) are as follows: the modified circumferential first-kind Mathieu functions
This article presents the outline of a set of routines that enable the computation of all Mathieu functions of integer orders for large range of the order n and the parameter h.
The accuracy of the results can be checked using the Wronskians for Mathieu functions, as the algorithms for computing Mathieu functions are not based on the Wronskian of the functions.
Once Mathieu coefficients of the appropriate kind, type, and order have been computed, the computations of the Mathieu functions are straightforward.
The functions MathuZn and MathuMZn are used to compute the radial and Modified radial Mathieu functions respectively.
Figures 4-5 show the modified first- and second-kind radial Mathieu functions for n = 4, and Tables I and II show the accuracy for the modified even and odd radial Mathieu functions using the Wranskian.

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