I am looking for a way to solve the integral equations \begin{align} \int_0^\infty \operatorname{Re} \left\{ K(r,z,t) \right\}g_\pm(t) \, \mathrm{d} t &= h_\pm(r,z) \, , \\ \int_0^\infty \operatorname{Im} \left\{ K(r,z,t) \right\}f_\pm(t) \, \mathrm{d} t &= h_\pm(r,z) \, , \end{align} wherein $$ K(r,z,t) = \left( r^2 + (z-it)^2 \right)^{-\frac{1}{2}} $$ is the kernel function and the right-hand side is given by $$ h_\pm(r,z) = \tfrac{\pi}{2} \left( r^2 + (z\pm 1)^2 \right)^{-\frac{1}{2}} \, . $$
Here, $(r,z) \in \mathbb{R}_+^2$. Using the method of residues, it can easily be shown that $$ f_+(t) = \frac{t}{1+t^2} \, , \qquad g_+(t) = \frac{1}{1+t^2} \, . $$
However, the task seems to be more delicate for $f_-(t)$ and $g_-(t)$. It would be great if someone could be of help and clarify how this problem can be tackled.
Hint: it looks like applying Hankel transforms on both sides of the equations render the integral equations a bit simpler.