# Möbius transformation from the complement of the closure of a disc to the unit disc

Let $D$ be the unit disc. Let $U$ be a disc of radius $R$ centered at some $\alpha \in \mathbb{C}$. I want to show that there is a Möbius transformation from $\mathbb{C_{\infty}}$\ $\bar{U}$ into $D$.

Here's what I've tried to do: Knowing that every Möbius transformation is a composition of inversions, rotations, dilations and translations, one can first dilate the unit disc $D$ to a disc $U$ of radius $R$ then translate it so that it is centered at $\alpha$. Once there one can apply inversion so we can end up outside of the closure of $U$. But it seems like I did the reverse that what I wanted. How can I re-organize all that in a rigorous way so I can have a clean formula for the Möbius transformation? Thx.

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Here, finding a Möbius transformation sending the complement of $\overline{U}$ to $D$ will be the same as finding one that sends $U$ to the complement of $\overline{D}$. So find a way to map $U$ onto $D$ and then take the reciprocal. Writing down a formula should be very straightforward! –  Dylan Moreland Aug 7 '11 at 2:15

Apply your inversion first, then your dilations and translations. The standard inversion map, $z\mapsto 1/z$, inverts in the unit circle centred at 0 - in particular, this map swaps the inside and the outside of $D$.