Let $\epsilon>0$. I was asked to find a conformal mapping from
$(\mathbb{R}\times(0,2))-((-\infty,i-\epsilon ] \cup[i+\epsilon,i+\infty))$
(An infinite horizontal strip but chopped a fine strip symmetrically)
to $\mathbb{H}$, the upper half plane.
The main obstacle is that I do not know how to deal with the chopped part. Does inversion help? But this is just a bounded strip instead of a "unbouneded" strip.