Problem of the Week #301 - June 28, 2022

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In summary, the conversation was about the importance of staying informed and up-to-date on current events. The speaker emphasized the need to fact-check information and not rely solely on social media for news. They also discussed the impact of fake news and the responsibility of individuals to research and verify information before sharing it. Overall, the conversation highlighted the importance of critical thinking and being an informed citizen.
  • #1
Euge
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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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  • #2
Here is my 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.
 
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