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Thread: complex analysis

  1. #1
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    complex analysis

    Let $\displaystyle \gamma$: [0,1] $\displaystyle \rightarrow C$ be any $\displaystyle C^1$ curve. Define $\displaystyle f(z)=\oint_{\gamma} \frac{1}{\zeta-z}d\zeta$
    Prove that $\displaystyle f$ is holomorphic on $\displaystyle C$\$\displaystyle \tilde{\gamma}$, where $\displaystyle \tilde{\gamma}=\{ \gamma (t):0\leq t \leq 1\}.$
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  2. #2
    Super Member Rebesques's Avatar
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    You can compute the difference quotient $\displaystyle [f(z+w)-f(z)]/w$ as $\displaystyle w\rightarrow 0$. The limit is (what a surprise) $\displaystyle f'(z)=-\int_{\gamma} \frac{1}{(\zeta-z)^2}d\zeta$.



    ps. what's with the \oint? I believe that notation is,
    only to signify counter-clockwise integration along a closed curve
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