I am trying to solve below integration. I believe that residue method would be the best approach to solve the integration.
$$S=\int_0^{\infty} \frac{H^{(1)}_{0}(ak)H^{(2)}_{0}(bk)}{k-p}-\frac{H^{(2)}_{0}(ak)H^{(1)}_{0}(bk)}{k+p}dk$$
However, I have some difficulty to find a proper contour for this. Small help is highly appriciated!