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Let $U_r = \mathbb{C} \backslash (-\infty,r]$. Find a conformal map from $U_r$ to the unit disc.

Idea: I can conformally map $U_r$ to $U_0$ with the transformation $z\mapsto z-r$. So we can write every element in the form $z= r e^{i\theta}$ with $|\theta| < \pi$ (branch cut). Then I can apply the map $z \mapsto z^{1/2}$. This maps $U_0$ to right half plane which can be conformally mapped (via $z\mapsto e^{i\pi/2}z$) to the upper half plane. Then I can use the conformal map $F(z) = \frac{i-z}{i+z}$ to map the upper half plane to the disc. Is my idea correct or is there any mistake?

Thanks in advance!

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    $\begingroup$ Looks good to me $\endgroup$ – MPW Jun 7 '17 at 16:37
  • $\begingroup$ Okay thanks to check my idea! $\endgroup$ – bob Jun 7 '17 at 16:44

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