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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!

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  • $\begingroup$ Are you looking for an explicit formula? $\endgroup$ – R. Burton Dec 26 '18 at 15:04
  • $\begingroup$ Yes, an explicit formula. $\endgroup$ – Yuyi Zhang Dec 27 '18 at 0:18
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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.

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