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Thread: 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 $\displaystyle \Omega \subset \field{C}.$ and satisfies the equation $\displaystyle |f(z)|=5$ for all $\displaystyle z\in \Omega$. Determine the function f.

    I started by saying $\displaystyle f = u + iv$. We know $\displaystyle u^2+v^2 = 25$.
    And we know also the Cauchy-Riemann equations since f is differentiable. I tried taking derivatives in $\displaystyle u^2+v^2 = 25$ but I was stuck. An intuitive function to have is $\displaystyle f = 5c$ 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, $\displaystyle f$ must be constant in $\displaystyle \Omega$ so $\displaystyle f(z)=5e^{i\theta_0}$ with $\displaystyle \theta_0\in [0,2\pi)$ .
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