Problem 1600 (difficulty: 8/10)

Prove for any complex number \(\displaystyle a\) that

\(\displaystyle \frac1{2\pi} \int_0^{2\pi} \log\big|e^{it}+a\big| \dt = \begin{cases} \log|a| & \text{if $|a|>1$,} \\ 0 & \text{if $|a|\le1$.} \\ \end{cases} \)

Give me another random problem!