A cylinder of radius $a$ has axis that passes through the origin perpendicular to the $x-y$ plane. A line vortex of circulation $\Gamma$ has axis passing through $x=b,y=0, (b>a)$ also perpendicular to the $x-y$ plane.
I'm struggling to find the complex potential for this system.
I know that the complex potential for a line vortex at the origin is $$w(z)=\frac{-i\Gamma}{2\pi}\ln(z)$$ with $v_{R}=0$ and $v_{\theta}=\frac{\Gamma}{(2\pi R)}$.
I tried to apply The Milne-Thomson's circle theorem to the problem, i.e
$$w(z)=f(z)+\overline{f\left(\frac{a^{2}}{\bar{z}}\right)}$$ But I'm unsure what $f(z)$ should be from the wording of the question, is it $f(z)=-\frac{i\Gamma}{2\pi}\ln(z)$?