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Math Help - Holomorphic functions with constant modulus

  1. #1
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    Holomorphic functions with constant modulus

    Problem Statement: Suppose that f is holomorphic on a domain  \Omega \subseteq \mathbb{C} with |f| constant. Prove that f is a constant map.

    Ideas: If |f| = 0 then f = 0. Suppose |f| = b \in \mathbb{C}, b \ne 0. So f(z) = b e^{i \theta(z)} where \theta(z) = arg_0(f(z)) with arg_0(f(z)) is a holomorphic branch of arg(f(z)); if such a branch did not exist then f would not be holomorphic on \Omega. I tried applying the definition of the complex derivative directly to show that f' = 0 to no avail. If it were shown that for any z_1 \ne z_2 in \Omega that \theta(z_1) = \theta(z_2) it would be done, but I don't see how to make that work either.

    Any hints, ideas, or a solution are welcome at this point. Thank you.
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  2. #2
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    If you want a one-liner use the open mapping theorem, otherwise use the C-R equations on the identity u^2+v^2=|f|^2=c where f(z)=u(z)+iv(z).
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