Find an explicit conformal map from region $G=D/\left\{0\leq x<1\right\}$ onto the unit disc $D$.

I know that we can construct a conformal map from a half plane or a disk to a disk via fractional map and from strip to sectors via exponential functions.But I don't know how to deal with the $G=D/\left\{0\leq x<1\right\}$.

  • $\begingroup$ Is your region the slit disk ? $\endgroup$ – H. H. Rugh Aug 25 '16 at 22:33

The map $$z\mapsto w=\sqrt{z}$$ (slit along ${\Bbb R}_+$ values in the upper half plane) takes $G$ to a half disk in the upper half plane $\{w: |w|< 1, \mbox{im }w>0\}$ . Then $$ w\mapsto u=\frac{1+w}{1-w}$$ (recall: Möbius maps circles and lines to circles and lines) maps the halfdisk to the quarter plane $\{u: \mbox{re } u >0, \mbox{im }u>0\}$. Then $$ u \mapsto v=u^2$$ takes you into the upper half plane. Finally, $$ v\mapsto \frac{v-i}{v+i}$$ maps you onto the unit disk.

  • $\begingroup$ Shouldn't it be $w = \frac{z+1}{z-1}$? I thought $\sqrt{z}$ takes the disk slit along the positive real axis to a half disk in the upper half plane... $\endgroup$ – Futurologist Aug 25 '16 at 23:19
  • $\begingroup$ Thanks, I hope I have got it right this time. $\endgroup$ – H. H. Rugh Aug 25 '16 at 23:23

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