# Integral of Bessel function with general Gaussian

I am trying to solve an integral like

$F(q,\alpha)=\int_0^\infty dx~ x~ e^{-(x-\alpha)^2/\beta^2}J_0(qx)$ where $\alpha,\beta>0,\in \mathbb{R}$; $q\geq 0,\in \mathbb{R}$

For a similar result, it is well known that $\int_0^\infty dx~ x ~e^{-x^2/\beta^2}J_0(qx)=\frac{\beta^2}{2}e^{-q^2\beta^2/4}$. However, I am unable to find any standard tricks (like one does with a normal Gaussian when shifted) in order to find the closed form expression for $F(q,\alpha)$.

I am able to verify numerically that this integral exists for a given $q,\alpha$ on the intervals specified, and have tried to compute this symbolically in Mathematica to no avail. My first thought was to expand the exponential and then do the integral, where I find the Taylor expansion for the solution,

$F(q,\alpha)\approx -\frac{0.735759}{\alpha q^3} -\frac{2.20728}{\alpha^3 q^5}+\frac{5.51819}{\alpha^5 q^7} +\frac{373.398}{\alpha^7 q^9}+ \frac{9705.12}{\alpha^9 q^{11}} + \frac{229893.}{\alpha^{11} q^{13}} +\frac{5.37139\times 10^6}{\alpha^{13} q^{15}}+...$

(Note that these numerial coefficients are to high precision consistent with $-\frac{2}{e}$,$-\frac{6}{e}$,$\frac{15}{e}$,$\frac{1015}{e}$,...)

Then my thinking was that if I can guess the solution, which from the solved example above is potentially composed of exponetials, and confirm this by Taylor expand my guess. This program has been unsuccessful so far.

A proof is not necessary, maybe someone knows a reference where this is done, or a means to finding the solution. However, a closed form expression is what I am after, unfortunately the numerical solution is not sufficient.

• I'm not sure if they cover this specific formula in there, but I know A Treatise on Bessel Functions covers many similar relations – Grant B. Mar 15 '17 at 5:45
• Thanks for your response @GrantB. Unfortunately it seems they, like all texts I've found, only consider generalizations of integrals like $\int_0^\infty ~dt~ t ~e^{-at}J_0(t)$ or $\int_0^\infty~ dt~ t ~e^{-bt^2}J_0(t)$. In principle I just need an integral $\int_0^\infty ~dt~ t~ e^{-at} e^{-bt^2} J_0(t)$. – AltaSkiier Mar 15 '17 at 15:33