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Math Help - Complex analysis and Taylor development

  1. #1
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    Complex analysis and Taylor development

    Hey guys.
    I need to prove this equation using Taylor development.
    It's pretty obvious that if I'm only using the first part of the Taylor development (the part I marked in red) I can prove it, but my question is, is that enough? can I do something like that?

    Thanks a lot.
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  2. #2
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    If \gamma is a contour containing ia\not = 0 then:
    \frac{1}{2\pi i} \oint_{\gamma} \frac{e^{iz}}{z-ia} dz = f ' (ia) where f(z) = e^{iz} by Cauchy's theorem.
    Therefore,
    \frac{1}{2\pi i} \oint_{\gamma} \frac{e^{iz}}{z-ia}dz = i e^{-a} \implies \frac{1}{2a i}\oint_{\gamma} \frac{e^{iz}}{z-ia} dz = \frac{\pi i e^{-a}}{a}
    Thus, your equation seems to be wrong.
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