It can be shown using the theorem of residues that
$$ \left( r^2+(z+h)^2 \right)^{-\frac{1}{2}} = \operatorname{Im} \int_0^\infty \frac{2}{\pi}\frac{ f(t) \, \mathrm{d}t}{\left( r^2 + (z-it)^2 \right)^\frac{1}{2}} \, , $$ with $r, z, h \in \mathbb{R}_+$. The kernel function is explicitly given by $$ f(t) = \frac{t}{t^2+h^2} \, . $$
I was wondering whether an analogous integral formula can be written for $ \left( r^2+(z-h)^2 \right)^{-\frac{1}{2}} $ as well, always for $r,z,h \in \mathbb{R}_+$
Those integral representation are useful for my further mathematical analysis. Any help/hint/suggestion/idea are highly appreciated.
Remark:
It can be checked that the above integral representation applies for harmonic functions that satisfy Laplace equation. Specifically, in cylindrical coordinates, both sides of the equation satisfy: $$ \frac{1}{r} \frac{\partial}{\partial r} \left( r \, \frac{\partial}{\partial r} (\cdot)\right) + \frac{\partial^2}{\partial z^2} (\cdot) = 0 \, . $$
Thank you,