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Problem of the Week #269 - Apr 24, 2018

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Euge

MHB Global Moderator
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Jun 20, 2014
1,892
Here is this week's POTW:

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Suppose $f$ is analytic on a simple closed contour $c$ in the complex plane. Prove $\displaystyle\int_c \overline{f(z)}f’(z)\, dz$ is purely imaginary.


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Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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Euge

MHB Global Moderator
Staff member
Jun 20, 2014
1,892
No one answered this week’s problem. You can read my solution below.


The real part of the integral is
$$\frac{1}{2} \int_c \overline{f(z)}f’(z)\, dz + f(z)\overline{f’(z)}\, d\overline{z} = \frac{1}{2}\int_c \overline{f(z)}df(z) + f(z)d\overline{f(z)} = \frac{1}{2}\int_c d\lvert f\rvert^2 = 0$$
 
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