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.
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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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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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FAQ: Problem of the Week #301 - June 28, 2022

What is the "Problem of the Week #301"?

The "Problem of the Week #301" is a weekly challenge or puzzle that is released by a scientific organization or community. It typically involves a specific topic or problem related to a particular field of science.

When will "Problem of the Week #301" be released?

"Problem of the Week #301" will be released on June 28, 2022. It is released on a specific date each week and can be found on the website or social media pages of the organization or community hosting the challenge.

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The purpose of "Problem of the Week #301" is to engage and challenge individuals in scientific thinking and problem-solving. It also serves as a way to promote and share knowledge in a specific field of science and bring together a community of like-minded individuals.

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