Everybody hello,
The goal is to determine the non-trivial conditions between the real parameters $a$, $b$, and $c$, for which infinite integral below is convergent / defined: $$ \int_0^\infty q J_0(qr) \big( q(a+b\cos q) + c \sin q\big) \, \mathrm{d}q \, , $$ where $0 \le r \le 1$.
My search through the literature has brought me the following relations which might be of use:
$$ \int_0^\infty J_0(qr)\cos(q) \, \mathrm{d}q = \frac{H(r-1)}{(r^2-1)^{1/2}} \, , \\ \int_0^\infty J_0(qr)\sin(q) \, \mathrm{d}q = \frac{H(1-r)}{(1-r^2)^{1/2}} \, , $$ where $H$ denotes the Heaviside step function.
Any hints or suggestions are most welcome. Thank you!