×

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 \(F(u)=\int^{\infty}_{a}f(x)\psi (x,u)dx,\) where \(\psi\) (x,u) denotes the Mathieu function of the third kind \(M_ r^{(3)}(x+i\pi)\) which satisfies the modified form of Mathieu’s equation \(\psi_{xx}=(u^ 2+2h^ 2 \cosh 2x)\psi,\) 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 \(e^{-\lambda x}f(x)\in L^ 2(a,\infty)\) where \(\lambda\geq 0\). An explicit eigenfunction expansion is obtainable for the case \(\lambda =0\).
Reviewer: D.Naylor

MSC:

44A15 Special integral transforms (Legendre, Hilbert, etc.)
34L99 Ordinary differential operators
33E10 Lamé, Mathieu, and spheroidal wave functions
PDFBibTeX XMLCite
Full Text: DOI