Exact expansions of Hankel transforms and related integrals

Abstract

The Hankel transform H n [f (x)](q) = ∞ 0 xf (x)J n (qx)dx is studied for integer n −1 and positive parameter q. It is proved that the Hankel transform is given by uniformly and absolutely convergent series in reciprocal powers of q, provided special conditions on the function f (x) and its derivatives are imposed. It is necessary to underline that similar formulas obtained previously are in fact asymptotic expansions only valid when q tends to infinity. If one of the conditions is violated, our series become asymptotic series. The validity of the formulas is illustrated by a number of examples.