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On an integral transform. (English) Zbl 0659.44004

The author obtains an inversion formula for the integral transform: \((*)\quad F(u)=\int^{a}_{0}J_ u(kr)f(r)dr/r\) where k, a are positive constants and u, the order of the Bessel function \(J_ u(x)\), is a complex parameter. The result is presented in the form of the following theorem: Suppose that f(r) is continuous for \(0<r\leq a\) and \(f(r)=O(r^ b)\) as \(r\to 0\), the integral transform (*) possesses the inverse: \[ f(r)=i\int_{L}(u\phi (u,r)F(u)du/J_ u(ka)), \] where \(0<r<a\), \(\phi (u,r)=J_ u(kr)Y_ u(ka)-J_ u(ka)Y_ u(kr)\) and the path of integration L in the complex u plane is the line \(Re(u)=c\) positioned so that \(c>-b\) and so that all the zeros of \(J_ u(ka)\) lie to the left of it.
Reviewer: K.N.Srivastava

MSC:

44A15 Special integral transforms (Legendre, Hilbert, etc.)
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