Evaluate the integral
$\int_{\gamma}e^{z^2}+ \overline{z} \ \ dz, $
where $\gamma$ is the positively oriented unit circle.
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Sign up to join this communityEvaluate the integral
$\int_{\gamma}e^{z^2}+ \overline{z} \ \ dz, $
where $\gamma$ is the positively oriented unit circle.
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 $dz=ie^{i\theta}d\theta$.