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Zbl 0685.44003
Naylor, D.
On an integral transform involving a class of Mathieu functions. (English)
[J] SIAM J. Math. Anal. 20, No.6, 1500-1513 (1989). ISSN 0036-1410; ISSN 1095-7154/e

The author develops an inversion formula associated with the integral transform F(u) defined by the equation $F(u)=\int\sp{\infty}\sb{a}f(x)\psi (x,u)dx,$ where $\psi$ (x,u) denotes the Mathieu function of the third kind $M\sb r\sp{(3)}(x+i\pi)$ which satisfies the modified form of Mathieu's equation $\psi\sb{xx}=(u\sp 2+2h\sp 2 \cosh 2x)\psi,$ 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 $e\sp{-\lambda x}f(x)\in L\sp 2(a,\infty)$ where $\lambda\ge 0$. An explicit eigenfunction expansion is obtainable for the case $\lambda =0$.
[D.Naylor]

MSC 2000:: *44A15 Special transforms
34L99 Ordinary differential operators
33E10 Spheroidal wave functions, etc.

Keywords: inversion formula; Mathieu function; Mathieu's equation; eigenfunction expansion

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