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I want to find a comformal mapping $\varphi$ which maps of domain $\{z\in\mathbb{C}:0<\Im(z)<\pi,|z|>r\},0<r<\pi$ onto the stip domain $\{z\in\mathbb{C}:0<\Im(z)<\pi\}$.

This problem is very important to me, I will be very appreciated if you could help me. Thank you very much.

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  • $\begingroup$ So to clarify, you want to map the upper half plane without the interior of the semicircle of radius $\pi$ onto the upper half plane? Also, I'm not familiar with the $\frak{J}$ notation, this is just principle argument correct? $\endgroup$ – Brandon Thomas Van Over Aug 20 '16 at 3:02
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First you divide r to get the unit, then square the original region get the whole space except an disk, then you inverse this function, so you get this disk. Finally, we use the conformal mapping from the disk to upper plane get the result. So $/phi(x)=i\frac{1+(\frac{r^2}{(f(z)^2)})}{1-(\frac{r^2}{(f(z)^2)})}$

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