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The standard form of the Mathieu differential equation is d2ydη2+(a−2qcos2η)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: (i)Ce1'(ξi,γ12)Ce1(ξi,γ12)(ii)Fey1'(ξi,γ12)Fey1(ξi,γ12)(iii)Gey1'(ξi,γ12)Gey1(ξi,γ12)(iv)Ce1'(ξi,γ22)Ce1(ξi,γ22)(iv)Se1'(ξi,γ22)Se1(ξi,γ22). 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

Mathieu functions and its useful approximation for elliptical waveguides