# 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

Consider the contrapositive of Proposition 6.4 which says If $$u$$ is not harmonic or $$v$$ is not harmonic, then $$f=u+iv$$ is not holomorphic in the region G.
Let $$u(x,y) = x^{3}+y^{3}$$. Then $$u_{xx}= (3x^{2})_x=6x$$ and $$u_{yy}=(3y^{2})_y=6y$$. So $$u_{xx}+u_{yy}=6x+6y$$ and we can choose values for $$x$$ and $$y$$ such that $$u_{xx}+u_{yy}\neq0$$. Thus, $$x^{3}+y^{3}$$ does not satisfy the Laplace equation $$u_{xx}+u_{yy}=0$$ for all $$(x,y)$$ and is therefore not harmonic. Since $$u(x,y)$$ is not harmonic, $$f=u+iv$$ is not holomorphic by the contrapositive of Proposition 6.4. So no matter what $$v(x,y)$$ we choose, $$f$$ will never be harmonic because we already know that $$u(x,y)$$ is not harmonic.