The asymptotic expansions of Hankel transforms and related integrals

In this paper, the asymptotic expansion of integrals of the form $\int_0^\infty {F(kr)f(k)dk}$ s considered, as $r$ tends to infinity, and where $F(kr)$ are Bessel functions of the first and second kind, or functions closely related to these. Asymptotic expansions for several functions of this type are presented under suitable restrictions on $f(k)$. The expansion given by Willis for Hankel transforms is seen to be valid under conditions of $f(k)$ less restrictive than those imposed by that author.