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Math Help - Complex Anaylsis: NEED HELP

  1. #1
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    Complex Anaylsis: NEED HELP

    I have 2 questions regarding complex analysis:

    1) Let f be analytic inside and on a closed contour gamma. Show that

    the integral of [f'(z)/(z-w)] dz = the integral of [f(z)/(z-w)^2] dz

    for all w not on Gamma.

    2) Explain why f(z) = e^z^2 has an antiderivative in the whole plane.
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  2. #2
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    Quote Originally Posted by Vitava61 View Post
    1) Let f be analytic inside and on a closed contour gamma. Show that

    the integral of [f'(z)/(z-w)] dz = the integral of [f(z)/(z-w)^2] dz
    Since f is analytic on a simply connected set containing \Gamma it means,
    \frac{1}{2\pi i}\oint_{\Gamma}\frac{f(z)}{(z-w)^2} = f''(w)
    And that,
    \frac{1}{2\pi i}\oint_{\Gamma} \frac{f'(z)}{(z-w)} = [f'(z)]'|_w = f''(w)
    Both follow by the Cauchy Integral Formula.
    2) Explain why f(z) = e^z^2 has an antiderivative in the whole plane.
    Define F(z) = \int_{[0,z]}e^{w^2} dw.
    Where [0,z] is the line segment connecting 0 to z.
    Then, F'(z) = e^{z^2}.
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