Naylor, D. On an integral transform involving a class of Mathieu functions. (English) Zbl 0685.44003 SIAM J. Math. Anal. 20, No. 6, 1500-1513 (1989). 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 Cited in 1 Document MSC: 44A15 Special integral transforms (Legendre, Hilbert, etc.) 34L99 Ordinary differential operators 33E10 Lamé, Mathieu, and spheroidal wave functions Keywords:inversion formula; Mathieu function; Mathieu’s equation; eigenfunction expansion PDFBibTeX XMLCite \textit{D. Naylor}, SIAM J. Math. Anal. 20, No. 6, 1500--1513 (1989; Zbl 0685.44003) Full Text: DOI Digital Library of Mathematical Functions: §28.26(ii) Uniform Approximations ‣ §28.26 Asymptotic Approximations for Large 𝑞 ‣ Modified Mathieu Functions ‣ Chapter 28 Mathieu Functions and Hill’s Equation