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Thread: Rectangle Theorem

  1. #1
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    Rectangle Theorem

    Theorem. Suppose $\displaystyle f $ is entire and $\displaystyle \Gamma $ is the boundary of a rectangle $\displaystyle R $. Then $\displaystyle \int_{\Gamma} f(z) \ dz = 0 $.

    Are we assuming that the rectangle bounds the curve? Or the curve bounds the rectangle?

    So the general idea of the proof is that we bisect a rectangle infinitely many times? Because then we have $\displaystyle \int_{\Gamma} f = \sum_{i=1}^{4} \int_{\Gamma_{i}} f $. Then why is it that from some $\displaystyle \Gamma_k, \ 1 \leq k \leq 4 $ we have $\displaystyle \int_{\Gamma^{(1)}} f(z) \ dz \gg I/4 $?
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    Quote Originally Posted by manjohn12 View Post
    Theorem. Suppose $\displaystyle f $ is entire and $\displaystyle \Gamma $ is the boundary of a rectangle $\displaystyle R $. Then $\displaystyle \int_{\Gamma} f(z) \ dz = 0 $.

    Are we assuming that the rectangle bounds the curve? Or the curve bounds the rectangle?

    So the general idea of the proof is that we bisect a rectangle infinitely many times? Because then we have $\displaystyle \int_{\Gamma} f = \sum_{i=1}^{4} \int_{\Gamma_{i}} f $. Then why is it that from some $\displaystyle \Gamma_k, \ 1 \leq k \leq 4 $ we have $\displaystyle \int_{\Gamma^{(1)}} f(z) \ dz \gg I/4 $?
    The meaning of $\displaystyle \oint_{\Gamma} f(z) dz = 0$ here is that you parametrize the rectangle and then integrate over this curve.

    Here "rectangle" means a figure consisting of sides made out of lines which are parallel and perpendicular to the x-y axes. "Rectangle" here does not mean the interior of the rectangle. Remember contour integration is over curves.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    The meaning of $\displaystyle \oint_{\Gamma} f(z) dz = 0$ here is that you parametrize the rectangle and then integrate over this curve.

    Here "rectangle" means a figure consisting of sides made out of lines which are parallel and perpendicular to the x-y axes. "Rectangle" here does not mean the interior of the rectangle. Remember contour integration is over curves.
    So suppose you have a curve $\displaystyle C_1 $ and you take $\displaystyle \int_{C_1} f(z) \ dz $. We can find the boundary of a rectangle $\displaystyle \Gamma_1 $ such that $\displaystyle \int_{C_1} f(z) \ dz = \int_{\Gamma_1} f(z) \ dz $? E.g. the boundary of the rectangle "contains" $\displaystyle C_1 $?
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