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Math Help - Finding a Complex function

  1. #1
    Member mohammadfawaz's Avatar
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    Finding a Complex function

    Hello,
    I have the following question:
    A complex function f is differentiable in a region \Omega \subset \field{C}. and satisfies the equation |f(z)|=5 for all z\in \Omega. Determine the function f.

    I started by saying f = u + iv. We know u^2+v^2 = 25.
    And we know also the Cauchy-Riemann equations since f is differentiable. I tried taking derivatives in u^2+v^2 = 25 but I was stuck. An intuitive function to have is f = 5c where c is a complex number with magnitude 1.

    Any help is appreciated.

    Thanx
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    According to the Maximum Modulus Principle, f must be constant in \Omega so f(z)=5e^{i\theta_0} with \theta_0\in [0,2\pi) .
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