# Finding a conformal map from the exterior of unit disk onto the exterior of an ellipse

Find a conformal bijection $f(z):\mathbb{C}\setminus D\rightarrow \mathbb{C}\setminus E(a, b)$ where $E(a, b)$ is the ellipse $\{x + iy : \frac{x^2}{a}+\frac{y^2}{b}\leq1\}$

Here $D$ denotes the closed unit disk.

I hate to ask having not given the question a significant amount of thought, but due to illness I missed several classes, really need to catch up, and the text book we're using (Ahlfors) doesn't seem to have anything on the mapping of circles to ellipses except a discussion of level curves on pages 94-95. and I can't figure out how to get there through composition of the normal elementary maps (powers, exponential and logarithmic), and fractional linear transformations take circles into circles and are therefore useless for figuring this out.

I prefer hints, thanks in advance.

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$z \mapsto \tfrac{1}{2} \left( z + \tfrac{1}{z} \right)$ maps $\{ z \mid |z| > R \}$ for each $R > 0$ biholomorphically to the exterior of some ellipse (depending on $R$) with foci $z = \pm 1$. Modify this map based on the foci of the given ellipse. – WimC Nov 13 '12 at 5:45

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