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Math Help - Schwarz reflection principle

  1. #1
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    Schwarz reflection principle

    Hi for all
    can u help me in solving this,or at least Give me a hint

    Applications to Schwarz reflection principle ; Show that there is no non-constant holomorphic function in the unit disc which is real- valued on the unit circle.
    Thanks
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  2. #2
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    Quote Originally Posted by raed View Post
    Hi for all
    can u help me in solving this,or at least Give me a hint

    Applications to Schwarz reflection principle ; Show that there is no non-constant holomorphic function in the unit disc which is real- valued on the unit circle.
    Thanks
    Transfer the holomorphic function f from the unit disc to the upper half-plane by the conformal map z = \frac{w-i}{w+i} (so that f(z) becomes F(w)). The boundary of the disc (the unit circle) gets transformed to the real axis in the w-plane, and you can then use the Schwarz principle to conclude that F is constant and hence so is f.
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  3. #3
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    Thanks alot

    Quote Originally Posted by Opalg View Post
    Transfer the holomorphic function f from the unit disc to the upper half-plane by the conformal map z = \frac{w-i}{w+i} (so that f(z) becomes F(w)). The boundary of the disc (the unit circle) gets transformed to the real axis in the w-plane, and you can then use the Schwarz principle to conclude that F is constant and hence so is f.
    Thank u very much
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