The standard form of the Mathieu differential equation is (d~2y)/(dη~2) + (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) (Ce'_1(ξ_i,γ_1~2))/(Ce_1(ξ_i,γ_1~2)) (ii) (Fey'_1(ξ_i,γ_1~2))/(Fey_1(ξ_i,γ_1~2)) (iii) (Gey'_1(ξ_i,γ_1~2))/(Gey_1(ξ_i,γ_1~2)) (iv) (Ce'_1(ξ_i,-γ_2~2))/(Ce_1(ξ_i,-γ_2~2)) (v) (Se'_1(ξ_i,-γ_2~2))/(Se_1(ξ_i,-γ_2~2)). Let ξ_0 = ξ_i, 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.
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