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Has anyone any general tips for finding conformal tranformations from a domain to another? For example from $D=\{ z=x+iy | x^2+y^2<1, x^2-x+y^2>0 \}$ to the unit disc. This set is a disc minus a smaller disc, so what should i imagine to try? If someone could provide his reasoning step by step telling what ideas make him use certain transformations (also providing some variants, for example: 'i'd do this because, while if there had been this i would have done that') i would be very grateful, because i don't seem to get a general idea of what is happening. Thanks

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Looks like the disks are internally tangent at $z=1$. This looks like a special point for this domain. It's a good idea to map the special point to $\infty$ by a fractional linear transformation. This makes both circles a special kind of circles: namely, lines. And the domain becomes a strip.

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