Problem 1802 (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 \(\displaystyle f\).

\(\displaystyle \quad (x,y)\mapsto y; \quad xy; \quad x^3y+xy^3; \quad x^3y-xy^3 \)

Give me another random problem!