Proposition

Is it possible to find a real function $v(x, y)$ so that $x^3 + y^3 + iv(x, y)$ is holomorphic?

Solution

By Proposition 6.3 [A first course in complex analysis], the real and imaginary parts of a holomorphic function must be harmonic. $u(x, y) = x^3 + y^3$ is not harmonic because $u_{xx} + u_{yy} = 6(x + y) \ne 0$ when $x = y = 1$.