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Thread: Complex Analysis

  1. #1
    MHF Contributor chiph588@'s Avatar
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    Complex Analysis

    1. If $\displaystyle f $ is analytic in a closed bounded region $\displaystyle G $ and $\displaystyle f(z) \neq 0 $ in $\displaystyle G $, show
    that $\displaystyle |f| $ assumes its minimum value on the boundary of $\displaystyle G $.
    Hint: consider $\displaystyle \frac{1}{f} $.

    2. Use Problem 1 to prove the Fundamental Theorem of Algebra.
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  2. #2
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    Quote Originally Posted by chiph588@ View Post
    1. If $\displaystyle f $ is analytic in a closed bounded region $\displaystyle G $ and $\displaystyle f(z) \neq 0 $ in $\displaystyle G $, show
    that $\displaystyle |f| $ assumes its minimum value on the boundary of $\displaystyle G $.
    Hint: consider $\displaystyle \frac{1}{f} $.
    The minimum value of $\displaystyle |f|$ is the maximum value of $\displaystyle 1/|f|$.
    But $\displaystyle 1/|f|$ assumes it maximum on the boundary by the maximum modulos principle.
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    MHF Contributor chiph588@'s Avatar
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