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Find all analytic function $f: \mathbb C \rightarrow \mathbb C$ such that $|f^`(z)|$ constant on curves of the form $Ref$ constant.

This is one of the past comp question. Seriously I do not know where to start. I do not even understand what the question is asking here. I have difficulty understanding what kind of curve has the form $Ref$ constant (example please). When I need to find entire function in other problem, I usually think of Liouville as a rescue but this time I don't think Louiville is going to save me. Any rigorous solution will be much appreciated.

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The way I read it is that for a particular function f you choose an z0 and get f(z0). Then starting at z0 you construct the curve of the z such that Re f(z) = Re f(z0). The property you want is that on each such curve, |f'(z)| = |f'(z0)|. I don't know how to solve the problem of creating such a function other than constant or similar moderately trivial functions. – marty cohen Dec 31 '12 at 4:28
One approach exploits the answers to the closely related question at consider the real part of the inverse of $f'$. – whuber Dec 31 '12 at 19:51

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