# Find a conformal map from the unit disc minus a disc in its interior to the unit disc.

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.

• Anything even remotely holomorphic won't work, and I have little experience with non-holomorphic conformal maps – Hagen von Eitzen Jul 24 at 22:39
• But conformal maps are bijective holomorphic functions. – user8513188 Jul 24 at 22:53

$$\tan\left(\frac{\pi}{z+1}-\frac{3\pi}{4}\right)$$
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.