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Thread: Complex Analysis - f is holomorphic...

  1. #1
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    Complex Analysis - f is holomorphic...

    Hey guys. Heres my problem:

    $\displaystyle f$ is holomorphic on $\displaystyle C$ and of the form $\displaystyle f(x + iy) = u(x) + i v(y)$ where $\displaystyle u$ and $\displaystyle v$ are real functions, then $\displaystyle f(x) = \lambda z + c$ with $\displaystyle \lambda \in R$ and $\displaystyle c \in C$.

    Where $\displaystyle C$ is the coplex numbers and $\displaystyle R$ is the real numbers.

    Earlier I've made an assignment where I show that if $\displaystyle f$ is holomorphic in a domain $\displaystyle G$ and $\displaystyle |f|$ is constant, then $\displaystyle f$ is constant. So I'm guessing I have to show that $\displaystyle f'$ is constant, which then tells me that there exists some $\displaystyle \lambda$ so $\displaystyle f(z) = \lambda z + c$, but I dont know how to do that.

    If anyone could assist I would greatly appreciate it.

    /Morten
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Complex Analysis - f is holomorphic...

    Use the Cauchy-Riemann equations to show that u' and v' are constant.
    Thanks from Deveno
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  3. #3
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    Re: Complex Analysis - f is holomorphic...

    Ah of course

    Thanks for the help.

    /Morten
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