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Math Help - Analytic Function in a domain D

  1. #1
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    Analytic Function in a domain D

    If f(z) is analytic in a domain D and maps D onto a portion of a straight line, then show that f(z) is a constant function in D.

    Anyone know how to solve this?
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  2. #2
    Super Member girdav's Avatar
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    Re: Analytic Function in a domain D

    Do you know the open mapping theorem?
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  3. #3
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    Re: Analytic Function in a domain D

    Don't really because i just start to learn complex analysis.
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  4. #4
    Super Member girdav's Avatar
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    Re: Analytic Function in a domain D

    In fact the open mapping theorem was a sledgehammer proof. We can write $f(z)=e^{i\theta z}a(z)$, where $\theta is a fixed real number. So g(z):=e^{-i\theta}f(z) is analytic on the open unit disk, and take only real values. By Cauchy-Riemann equations, what can you conclude?
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