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could any one help me to show that any $f$ analytic in the annulus $1\le |z|\le 2$ and $|f|$ is a constant on $|z|=1$ and on $|z|=2$ (not necessarily the same constant) admits a meromorphic extension to $\mathbb{C}^*$

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up vote 5 down vote accepted

The extension is given by repeated analytic reflection. Reflection (inversion) in the circle of radius $R$ is given by $\tau_R(z) = R^2/\bar{z}$. So if $|f(z)|=R$ for $|z|=2$, you can define $f(z) = \tau_R \circ f \circ \tau_2 (z)$ for $2<|z|\le 4$. (The reflected images of zeros become poles, so instead of an analytic function you only get a meromorphic function in general.) Repeated reflection in the outer circle gives a meromorphic map in $|z| \ge 1$. Reflection in the unit circle then gives you a meromorphic map in the punctured plane.

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