Need help with the specific question related to the circle theorem.
Let $z$ be a complex number. Show that the image of a line vortex of strength $2 \pi \kappa$ located at $z = b$ inside the circle $|z| = a $ can be written as:
$w_2(z) = i \kappa \log(z − a^2/b)$
I know that the line vortex inside the circle has the potential $w_1(z) = i \kappa \log(z − b)$ but it has a singularity at $z=b$ which is inside the circle so I cant apply the circle theorem.
I've also tried applying the circle theorem to $w_2(z)$ instead in hopes to find $w_1(z)$ but with little success
Any help would be great.