Problem of the Week #301 - June 28, 2022

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Here is this week's problem!

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Prove that if $f$ is holomorphic in an open subset $\Omega\subset \mathbb{C}$, then for all closed countours $\Gamma$ in $\Omega$, the integral $\oint_{\Gamma} \overline{f(z)}f’(z)\, dz$ is purely imaginary.
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Here is my solution.

Spoiler: Solution

The differential form $\overline{f(z)}f'(z)\, dz = \overline{f(z)}\,df(z)$, so $2\operatorname{Re}(\overline{f(z)}f'(z)\, dz) = \overline{f}\, df + f\, d\overline{f} = d(f\overline{f})$, an exact differential. Therefore $$2\operatorname{Re} \oint_\Gamma \overline{f(z)}f'(z)\, dz = \oint_\Gamma 2\operatorname{Re}(\overline{f(z)}f'(z)\, dz) = 0$$ and the result follows.