Problem 1799 (difficulty: 3/10)

Which ones of the following functions are harmonic?

For each harmonic function \(\displaystyle f\) from the list, provide a holomorphic function those real part is \(\displaystyle f\), and the harmonic conjugate of \(\displaystyle f\).

\(\displaystyle (x,y)\mapsto x; \quad (x,y)\mapsto x^2; \quad x^2+y^2; \quad x^2-y^2; \quad \log(x^2+y^2) \quad \frac{1-x^2-y^2}{1-2x+x^2+y^2} \)

Give me another random problem!