I have the following integral $$\int_{x=0}^\infty \frac{x^{-\frac{1}{2}}}{x^2+a^2} \, \mathrm{J}_{\frac{3}{2}}(b \, x) \, \mathrm{J}_{0}(c \, x) \, \mathrm{d}x\ $$
with $a^2 \in \mathbb{C}$, $\, b>0$ and $c>0$.
I can solve the integral for $c>b$, any ideas how to solve it for $c<b$.
Thank you.
Edit: The solution for the first case can be given with the general formula $$\int_{x=0}^\infty \frac{x^{\mu-\nu+2\ell+1}}{x^2+a^2} \, \mathrm{J}_{\mu}(b \, x) \, \mathrm{J}_{\nu}(c \, x) \, \mathrm{d}x\ = (-1)^\ell \, a^{\mu-\nu+2\ell} \, \mathrm{K}_{\mu}(b \, a) \, \mathrm{I}_{\nu}(c \, a)$$ with $-(\ell+1)<\mathrm{Re}(\mu)<\mathrm{Re}(\nu)-2\ell+2$ and $b>c$.
Your integral can be written as
$$I(a,b,c)=\sqrt{\frac{2}{\pi b^3}}\int_{0}^{+\infty}\frac{-bx\cos(bx)+\sin(bx)}{x^2(a^2+x^2)}\, J_0(cx)\,dx \tag{1}$$ and the last integral can be computed by switching to Fourier transforms: the Fourier transform of $J_0(cx)$ is a multiple of $\frac{1}{\sqrt{c^2-s^2}}$ supported on $(-c,c)$ and the Fourier transform of the other term is tedious but straightforward to compute.