# Moebius transformations mapping a doubly connected domain to an annulus

What are the Moebius transformations mapping the unbounded doubly connected domain in the extended complex plane, whose boundary consists of the circles $C_1:=\lbrace z\in\mathbb{C}:|z-5|=4\rbrace$ and $C_2:=\lbrace z\in\mathbb{C}:|z+5|=4\rbrace$, onto some annulus $\lbrace w\in\mathbb{C} : 1<|w|<R\rbrace$ ?

I also want to know whether any arbitrary multiply connected domains can be mapped to an annulus or not. If not what property should the domain satisfy in order to make this happen ?

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I wrote out a complete solution, but then thought that this smacks of homework, so let me just give a (strong!) hint:

Möbius transformations preserve circles (see Wikipedia, for example), so you need only map the centres of the two excised disks to zero and infinity, and then rescale so that the circle around zero has unit radius.

(If it's not homework, I'll happily fill in the details.)

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I can't make out your hint. Are you saying center will go to centre which is not true, why should one centre go to infinity ? – pritam Nov 6 '12 at 13:01
It is probably easiest to think about in terms of the Riemann sphere (Möbius transformations are one-to-one mappings of the Riemann sphere to itself). A disk at infinity projects to the outside of some circle. So you can think of an annulus as being obtained by removing two disks - one centred at zero, and one at infinity. – Rhys Nov 6 '12 at 13:18