I approached this integral to be solved:
$$\int_0^{+\infty}\,\frac{k}{k^2-\alpha^2}\,J_1(kR)\,J_1(ka)\,\exp(-d^2k^2)\,dk$$
where $J_1(x)$ is the Bessel function of the first kind and $R,a,d,\alpha$ have real positive values. The additional condition is that:
$$J_1(\alpha a)=0$$
i.e. $\alpha a$ is one of the positive zeros of $J_1(x)$.
From this, when $k\to\alpha$, the integrand is not divergent, because:
$$\lim_{k\to\alpha}\,\frac{J_1(ka)}{k^2-\alpha^2}=1$$
Any suggestion on how to solve this integral?