# Problem of the Week #301 - June 28, 2022

• MHB
• Euge
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.
Euge
Gold Member
MHB
POTW Director
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.

malawi_glenn

## 1. 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.

## 2. 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.

## 3. Who can participate in "Problem of the Week #301"?

Anyone with an interest in science and problem-solving can participate in "Problem of the Week #301". It is open to scientists, students, and the general public. Some challenges may have age or level restrictions, but most are open to all.

## 4. What is the purpose of "Problem of the Week #301"?

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.

## 5. How can I submit my solution for "Problem of the Week #301"?

The submission process for "Problem of the Week #301" may vary depending on the organization or community hosting the challenge. Typically, solutions can be submitted through a designated online form or by email. Make sure to carefully read the instructions and guidelines provided for each challenge.

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