By, of course, using the Definitions of those things. Writing z= x+ iy, |z|= so .

u(x,y) is harmonic if and only if it satisfies . Find those partial derivatives and add!

If is analytic then both u and v are harmonic functions and we say that they are "harmonic conjugates".

If f(z) is analytic it must satisfy and . Since you are given u, you can find the left sides of those and solve the resulting equations for v.