I am trying to solve an integral like
$$ F(q, a, b) = \int_0^{+\infty}dx e^{-(x-a)^2/b^2}J_0(qx)x \quad where\quad a, b > 0; q > 0 $$
For a similar result, it is well known that $\int_0^{+\infty}dx e^{-x^2/b^2}J_0(qx)x = \frac{b^2}{2}e^{-q^2 b^2 / 4}$. Is there a simple formula for this integral?
I search and find the same question in here, but no answer is available.