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Math Help - Complex Analysis-Maximum Principle

  1. #1
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    Complex Analysis-Maximum Principle

    Let f be an analytic function at the open unit circle and continous at |z|=1.

    Prove: If f(z)=1 on the upper half of the unit circle
    ( for  z=e^{i\gamma} where  0<= \gamma <=\pi
    then f is a constant at the unit circle...



    I've no idea about this question.., I'll be delighted to get some guidance....


    Thanks !
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  2. #2
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    Quote Originally Posted by WannaBe View Post
    Let f be an analytic function at the open unit circle and continous at |z|=1.

    Prove: If f(z)=1 on the upper half of the unit circle
    ( for  z=e^{i\gamma} where  0<= \gamma <=\pi
    then f is a constant at the unit circle...
    Using a conformal map from the unit disk to the upper half-plane that takes the upper part of the unit circle to the positive real axis, you can replace the problem by one in which g is an analytic function on the upper half-plane, continuous on the real axis and such that g(z)=1 on the positive real axis. Then apply the reflection principle, defining g(\overline{z}) = \overline{g(z)}, to extend g to an analytic function on the cut plane \mathbb{C}\setminus(-\infty,0]. This extended function is constant on the positive real axis and hence constant everywhere.
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  3. #3
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    Wow thanks a lot!
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