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$\int_{\gamma} ze^{-z} dz$ where ${\gamma}$ is the unit circle centered at the origin.

By Cauchy's Theorem it is the composition of functions analytic in C and so is analytic on and inside ${\gamma}$, therefore it is equal to 0.

But I'm looking for how you would answer this question using the FTC?

Edit: fixed the question

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    Fixed the question.2012-03-18

1 Answers 1

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Take any smooth parametrization $\gamma(t)$, $t\in[0,1]$. Then \int_{\gamma} ze^{-z} dz=\int_0^{1}\gamma(t)e^{-\gamma(t)}\gamma'(t)\,dt=\left.e^{-\gamma(t)}-\gamma(t)e^{-\gamma(t)}\right|_0^{1}=0 since $\gamma(0)=\gamma(1)$.

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    ok ty, i have to get my head around it.2012-03-19