Mathieu functions and its useful approximation for elliptical waveguides

Abstract

The standard form of the Mathieu differential equation is d2y /dη2 a aa2 cos2η) y 0 where a and q are real parameters and q > 0. In this paper we obtain closed formula for the generic term of expansions of modified Mathieu functions in terms of Bessel and modified Bessel functions in the following cases: (Equation presented) Let ξ0 = ξ0, where i can take the values 1 and 2 corresponding to the first and the second boundary. These approximations also provide alternative methods for numerical evaluation of Mathieu functions.