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First part of the question asks me to state the path integral $\int_\gamma f$, which I defined as: \begin{equation} \int_\gamma f = \int^b_a f(\gamma(t))\gamma ' (t) dt \end{equation}

And the second part asks to evaluate the integral \begin{equation} \int _\gamma (e{^z}^{2} + \overline{z}) dz \end{equation} where $\gamma$ is the positively oriented unit circle.

Does this mean that $\gamma(t) = e^t + t \ (0 \leq t \leq 1) $ ?

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  • $\begingroup$ $\gamma (t) = e^{2 \pi i t} \quad t \in (0,1)$, is a fairly standard parameterization of the unit circle $\endgroup$ – Triatticus May 27 '15 at 7:52
  • $\begingroup$ I think inside the integral you have $\;e^{z^2}\;$ ? $\endgroup$ – Timbuc May 27 '15 at 7:58
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Since $e^{z^2}$ is analytic on and inside the curve given and since $\overline{z}=1/z$ on this curve, the integral is $0 + 2\pi i$.

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