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I am trying to find a conformal map from the unit disc minus the disc, $|z+\frac{1}{2}|<\frac{1}{2}$, onto the unit disc. I've been playing around with linear fractional transformations, but I haven't been able to cook up anything useful yet. Any help is appreciated.

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  • $\begingroup$ Anything even remotely holomorphic won't work, and I have little experience with non-holomorphic conformal maps $\endgroup$ – Hagen von Eitzen Jul 24 at 22:39
  • $\begingroup$ But conformal maps are bijective holomorphic functions. $\endgroup$ – user8513188 Jul 24 at 22:53
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$$\tan\left(\frac{\pi}{z+1}-\frac{3\pi}{4}\right)$$

First $(z+1)^{-1}$ maps to the strip $\tfrac12 < \operatorname{Re}(z) < 1$. Then $\pi \,(z-\tfrac34)$ maps to the strip $-\pi/4 < \operatorname{Re}(z) < \pi/4$. Then $\tan(z)$ maps to the disc.

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