# Problem 6.8

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

Suppose we have the function $$f(x,y)=u(x,y) + iv(x,y)$$ such that $$u(x,y)= x^3 + y ^3.$$ If we take the contrapositive of Proposition 6.4 it states: If $$u$$ is not harmonic in G or $$v$$ is not harmonic in G. Then $$f=u+iv$$ is not holomorphic in the region G. Notice $$u_{xx}+ u_{yy}=6x + 6y$$, which is not equal to 0 when $$y\neq -x$$. So by Definition 6.1, $$u$$ is not harmonic for some $$x,y\in \mathbb{C}$$ But $$x, y$$ are not holomorophic on that line since it is impossible to have an open disk centered at $$z_0$$. Hence, $$u$$ is not holomorphic, for all $$x, y \in \mathbb{C}$$. Therefore, it is not possible to find a real function $$v(x,y)$$ so that $$x^3 + y^3 + iv(x,y)$$ is holomorphic.