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Math Help - Cauchy-Goursat Theorem

  1. #1
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    Cauchy-Goursat Theorem

    Prove the following by means of Cauchy-Goursat theorem. Begin with performed around |z|=1. Use the parametric representation Separate your equation into real and imaginary parts.
    12.
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  2. #2
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    You do not even need Cauchy closed curve theorem here. Just note that \left( e^z \right)' = e^z which means if \Gamma it a piecewise smooth closed curve then \oint_{\Gamma} e^z dz = 0.

    You prove this by writing,
    0=\oint \limits_{|z| = 1} e^z dz = \int_0^{2\pi} e^{e^{i\theta}} ie^{i\theta} d\theta \implies \int_0^{2\pi} e^{e^{i\theta}} e^{i\theta} = 0

    Now split the real and imaginary parts.
    Hint: e^{e^{i\theta}} = e^{\cos \theta} \cos (\sin \theta) + i e^{\cos \theta}\sin (\sin \theta).
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