Let $\mathbb{D}$ denotes the unit disk, then construct a conformal mapping that map the set
$S=\mathbb{D}$ \{(-1,$-\frac{1}{2}$]$\bigcup$[$\frac{1}{2}$,1)} onto $\mathbb{D}$ itself.
I know some basic knowledge about comformal mapping from disk onto disk. But when some parts of disk are digged out, I have no idea. Hope someone could help, thanks!
The Joukowsky transform $w$ maps the open unit disk $D$ to the complex plane minus a segment of the real axis. Further, $w$ maps $D \backslash ([-1, -1/2] \cup [1/2, 1])$ to the complex plane minus a wider segment of the real axis. Apply $w$ followed by a scaling and followed by $w^{-1}$ (with the correct choice of $w^{-1}$) to obtain the desired mapping.