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Is there a procedural way of finding a Möbius transformation given prescribed conditions?

For example, I've been asked to find a Möbius tranformation which fixes $\mathcal{C}_2$, maps $\mathcal{C}_1$ to a line parallel to the imaginary axis, and sends $4$ to $0$. (Here $\mathcal{C}_r$ denotes the complex circle with radius $r$, e.g. $\mathcal{C}_1$ is the unit circle.)

I'm less concerned about solving this particular problem as I am about figuring out if there is a more or less algorithmic way to go about problems of this type. I am aware that a Möbius transformation is completely defined by three points, but I'm not sure how to get this from the set images.

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2 Answers 2

Offhand, this seems like an over-determined problem. But there's some symmetry. Do you know what Möbius transformations map the unit disk to itself? Then can you generalize to the disk of radius $2$? Then think about how to get $4$ to map to $0$.

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I figured out that the Mobius transformations fixing the unit disc are $e^{i\theta}\frac{z+a}{\overline{a}z+1}$ but I can't figure out how to generalize to the disk of radius 2. –  Samuel Handwich Apr 22 '13 at 11:46
    
Good start. Presumably there's some condition on $|a|$ there that you need in order to guarantee that the interior maps to the interior. What if you want the exterior to map to the interior and vice-versa? ... Can't you now rescale the problem to get it to work for a disk of radius $2$? –  Ted Shifrin Apr 22 '13 at 20:35

Usually, the best way to solve these types of problems is by using the cross ratio

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