Exact expansions of Hankel transforms and related integrals

Abstract

The Hankel transform $\mathcal {H}_n [f(x)](q)=\int _0^{\infty } \!\! \, x f(x) J_n(q x) \mathrm{d}x$ is studied for integer $n\geqslant -1$ 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.

Appendix A

Here we calculate a series which defines integral (28). After using the relation

$$\begin{aligned} \Gamma (2m+2) = 2^{2m+1} \Gamma (m+1) \Gamma (m+3/2)/ \sqrt{\pi } , \end{aligned}$$

(A.1)

we find from (8) and (33)

$$\begin{aligned} H_3(q)&= \frac{2a}{\sqrt{\pi } q^3} \sum _{m=0}^\infty \frac{\Gamma (m+3/2)}{\Gamma (m+1)} \left( -\frac{a^2}{q^2}\right) ^{\!\!m} \sum _{k=0}^m \left( {\begin{array}{c}2m+1\\ 2k\end{array}}\right) \left( {\begin{array}{c}2k\\ k\end{array}}\right) \!\left( \frac{c^2}{4a^2}\right) ^{\!\!k} \nonumber \\&= \frac{2a}{\sqrt{\pi } q^3} \sum _{k=0}^\infty \frac{1}{(k!)^2} \left( \frac{c^2}{4a^2}\right) ^{\!\!k} \sum _{m=k}^\infty \frac{\Gamma (m+3/2)\,\Gamma (2m+2)}{\Gamma (m+1)\,\Gamma (2m+2-2k)} \left( -\frac{a^2}{q^2}\right) ^{\!\!m} . \end{aligned}$$

(A.2)

After putting $m=n+k$, we find

$$\begin{aligned} H_3(q)&=\frac{2a}{\sqrt{\pi } q^3} \sum _{k=0}^\infty \frac{1}{(k!)^2} \left( -\frac{c^2}{4q^2}\right) ^{\!\!k} \nonumber \\&\quad \times \sum _{n=0}^\infty \frac{\Gamma (n+k+3/2)\,\Gamma (2n+2k+2)}{\Gamma (n+k+1)\,\Gamma (2n+2)} \left( -\frac{a^2}{q^2}\right) ^{\!\!n} . \end{aligned}$$

(A.3)

Using relation

$$\begin{aligned} \frac{\Gamma (n+k+3/2)\,\Gamma (2n+2k+2)}{\Gamma (n+k+1)\,\Gamma (2n+2)} = 2^{2k} \, \frac{[\Gamma (n+k+3/2)]^2}{\Gamma (n+3/2) \, n!} , \end{aligned}$$

(A.4)

we obtain ($q > a+c$)

$$\begin{aligned} H_3(q)&= \frac{a}{q^3} \sum _{k=0}^\infty \frac{1}{(1)_k k!} \left( -\frac{c^2}{q^2}\right) ^{\!\!k} \sum _{n=0}^\infty \frac{[(3/2)_{n+k}]^2}{(3/2)_n n!} \left( -\frac{a^2}{q^2}\right) ^{\!\!n} \nonumber \\&= \frac{a}{q^3} \, F_4 \!\left( \frac{3}{2}, \frac{3}{2}, \frac{3}{2}, 1, -\frac{a^2}{q^2}, -\frac{c^2}{q^2} \right) , \end{aligned}$$

(A.5)

where

$$\begin{aligned} F_4(\alpha , \beta , \gamma , \gamma ', x, y) = \sum _{n.m=0}^\infty \frac{(\alpha )_{m+n} (\beta )_{m+n}}{(\gamma )_{m} (\gamma ')_{n} \,m! \,n!} \, x^m y^n \end{aligned}$$

(A.6)

is the hypergeometric series of two variables [24]. It is the uniformly and absolutely convergence series for $\sqrt{|x|} + \sqrt{|y|} < 1$.