# How can I construct this conformal mapping?

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!

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

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.