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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.2012-12-31
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    One approach exploits the answers to the closely related question at http://math.stackexchange.com/questions/267514: consider the real part of the inverse of $f'$.2012-12-31

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First of all, unless $f$ is constant, there exists some $z_0$ with $f'(z_0) \ne 0$. Let $w_0 = f(z_0)$, and let $g$ be the local inverse of $f$ with $g(w_0) = z_0$ in some small disk. Then $|g'(w)|$ is constant (and non-zero) on horizontal lines, so there exists an analytic branch of $h(w) = \log g'(w)$ which then has constant real part on horizontal lines. It is a standard exercise using the Cauchy-Riemann equations that this implies that $h$ is constant, which in turn implies that both $g'$ and $f'$ are locally constant. By the identity principle this implies that $f'$ is constant in the plane, so $f(z) = az+b$ with some constants $a$ and $b$>