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Math Help - Complex Analysis

  1. #1
    Senior Member Dinkydoe's Avatar
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    Complex Analysis

    The Cauchy Integral formula states the following:

    Suppose U\in \mathbb{C} is an open set and f is holomorphic on U. Let z_0 \in U and r>0 such that \overline{D}(z_0,r)\subset U. Let \gamma:[0,1]\to U be the C^1 curve \gamma(t)=z_0+r\cos(2\pi t)+ir\sin(2\pi t). Then f(z)=\int_\gamma \frac{f(\zeta)}{\zeta-z}d\zeta for z\in D(z_0,r)

    Now I have to show the Cauchy-integral formula is valid if we only assume that F\in C^0(\overline{D}), with D some disc, and F is holomorphic on D

    I'm not sure if I understand what must be shown here exactly. Can someone clarify this a little?
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  2. #2
    Super Member
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    Quote Originally Posted by Dinkydoe View Post
    The Cauchy Integral formula states the following:

    Suppose U\in \mathbb{C} is an open set and f is holomorphic on U. Let z_0 \in U and r>0 such that \overline{D}(z_0,r)\subset U. Let \gamma:[0,1]\to U be the C^1 curve \gamma(t)=z_0+r\cos(2\pi t)+ir\sin(2\pi t). Then f(z)=\int_\gamma \frac{f(\zeta)}{\zeta-z}d\zeta for z\in D(z_0,r)

    Now I have to show the Cauchy-integral formula is valid if we only assume that F\in C^0(\overline{D}), with D some disc, and F is holomorphic on D

    I'm not sure if I understand what must be shown here exactly. Can someone clarify this a little?
    My approach would be something like this:

    Let g_t(z)= \frac{1}{2\pi i} \int_{\gamma _t } \frac{f(\zeta )}{\zeta -z} d\zeta with t\in (\frac{1}{2} ,1) and \gamma _t= te^{2\pi is} (assuming z_0=0) argue that \lim_{t\rightarrow 1} g_t(z) is the function you're given and since (by the usual Cauchy integral formula) g_t(z)=f(z) for all z\in \mathbb{D}_{\frac{1}{2} } by Riemann's theorem we would have that g_1(z)=f(z) for all z\in \mathbb{D}
    Last edited by Jose27; March 28th 2010 at 05:45 PM.
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  3. #3
    Senior Member Dinkydoe's Avatar
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    ok, I take it with \mathbb{D}_{\frac{1}{2}} you mean the open ball with radius r=1/2 around z_0=0.

    Thanks for your answer however what theorem do you refer to when you say : " Riemann's Theorem''
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