Is the Integral of an Analytic Function on a Closed Contour Purely Imaginary?

[Euge]

Here is this week's POTW:

-----
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.-----

Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!

 

[Euge]

No one answered this week’s problem. You can read my solution below.

Spoiler

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