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On an integral transform involving a class of Mathieu functions. (English) Zbl 0685.44003
The author develops an inversion formula associated with the integral transform F(u) defined by the equation F(u)= a f(x)ψ(x,u)dx, where ψ (x,u) denotes the Mathieu function of the third kind M r (3) (x+iπ) which satisfies the modified form of Mathieu’s equation ψ xx =(u 2 +2h 2 cosh2x)ψ, h being a positive constant. The basic inversion formula is expressed as an integral in the complex u-plane and applies for functions f(x) such that e -λx f(x)L 2 (a,) where λ0. An explicit eigenfunction expansion is obtainable for the case λ=0.
Reviewer: D.Naylor
MSC:
44A15Special transforms (Legendre, Hilbert, etc.)
34L99Ordinary differential operators
33E10Lamé, Mathieu, and spheroidal wave functions