# Complex analysis, evaluating a path integral

Evaluate the integral

$\int_{\gamma}e^{z^2}+ \overline{z} \ \ dz,$

where $\gamma$ is the positively oriented unit circle.

• What steps have you taken to solve the problem? Apr 3, 2013 at 16:14
• Im guessing the path is something like $\gamma (t)=e^{it}$ with $0\leq t \leq 2\pi$ Apr 3, 2013 at 16:14
• Have you tried that out? Apr 3, 2013 at 16:15
• Yeah but the i get stuck with an $e^{e^{2it}}$ term and im not sure where to go then? Apr 3, 2013 at 16:16
• Let $\gamma$ be a closed loop. $\gamma$ bounds a region $U$. If on a domain $D\supset \gamma\cup U$, $f$ is analytic, then$\int _{\gamma} f(z) dz=0$. Do you know this result?# Apr 3, 2013 at 16:28

Hint: $$\int_\gamma e^{z^2} + \overline{z} \; \mathrm{d}z= \int_\gamma e^{z^2} \; \mathrm{dz}+\int_{\gamma} \overline{z} \; \mathrm{d}z$$ For the first one use that $e^{z^2}$ is holomorphic (so what is the value of the integral)? for the second use a parametrisation of the unit circle.
Just cause i like it, you can use that for $|z|=1$ the following equlity holds $$\frac{1}{z}=\overline{z}$$
Hint: Set $z=e^{i\theta}$ for $0\leq\theta<2\pi$ and notice $dz=ie^{i\theta}d\theta$.