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Thread: Proof Cauchy-Goursat

  1. #1
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    Proof Cauchy-Goursat

    Cauchy-Goursat Theorem: If $\displaystyle f(z) $ is analytic inside and on a simple, closed piecewise smooth curve $\displaystyle C $, then $\displaystyle \oint_{C} f(z) \ dz = 0 $.

    In proving the "special case" where we use Greens Theorem and the Cauchy Riemann Equations, is assuming that $\displaystyle f $ is continuously differentiable the same thing as saying that $\displaystyle f $ has continuous partial derivatives?
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    Quote Originally Posted by manjohn12 View Post
    Cauchy-Goursat Theorem: If $\displaystyle f(z) $ is analytic inside and on a simple, closed piecewise smooth curve $\displaystyle C $, then $\displaystyle \oint_{C} f(z) \ dz = 0 $.

    In proving the "special case" where we use Greens Theorem and the Cauchy Riemann Equations, is assuming that $\displaystyle f $ is continuously differentiable the same thing as saying that $\displaystyle f $ has continuous partial derivatives?
    If $\displaystyle f$ is continous differenciable (on some open set) it means $\displaystyle f$ is differenciable and $\displaystyle f'$ is continous (on this open set). Write $\displaystyle f = g + hi$. Remember that $\displaystyle f ' = g_x + h_x i$. Thus, $\displaystyle g_x,h_x$ are continous on this open set. However, by Cauchy-Riemann equations $\displaystyle g_x=h_y$ and $\displaystyle h_x = -g_y$ so $\displaystyle h_y,g_y$ are continous also. Thus, the partial derivatives must be continous.
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    Quote Originally Posted by ThePerfectHacker View Post
    If $\displaystyle f$ is continous differenciable (on some open set) it means $\displaystyle f$ is differenciable and $\displaystyle f'$ is continous (on this open set). Write $\displaystyle f = g + hi$. Remember that $\displaystyle f ' = g_x + h_x i$. Thus, $\displaystyle g_x,h_x$ are continous on this open set. However, by Cauchy-Riemann equations $\displaystyle g_x=h_y$ and $\displaystyle h_x = -g_y$ so $\displaystyle h_y,g_y$ are continous also. Thus, the partial derivatives must be continous.
    You mean $\displaystyle f_x = g_x +ih_x $?
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  4. #4
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    Quote Originally Posted by manjohn12 View Post
    You mean $\displaystyle f_x = g_x +ih_x $?
    No. The derivative of $\displaystyle f$ is computed to be $\displaystyle g_x + ih_x$.
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