Question: are all analytic functions harmonic?
Answer: If f(z) = u(x, y) + iv(x, y) is analytic on a region A then both u and v are harmonic functions on A. Proof. This is a simple consequence of the Cauchy-Riemann equations. ... To complete the tight connection between analytic and harmonic functions we show that any harmonic function is the real part of an analytic function.