Exact expansions of Hankel transforms and related integrals

TL;DR

This paper derives exact, convergent series expansions for the Hankel transform for integer orders, clarifying conditions under which these are valid and contrasting them with asymptotic series for large q.

Contribution

It provides new exact series expansions for the Hankel transform that are uniformly and absolutely convergent under certain conditions, improving upon previous asymptotic formulas.

Findings

Derived exact series expansions for Hankel transforms.

Identified conditions for convergence versus asymptotic behavior.

Validated formulas through multiple examples.

Abstract

The Hankel transform H_n[f(x)](q) = int_0^infinity 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.