Let $u(x,y)=x^3-3xy^2-3x^2y+y^3+x^2-y^2+2xy$. To prove that $u$ is harmonic, is showing that $\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2u}{\partial y^2}=0$ enough to show its harmonic?
Secondly, to find a harmonic function $v:\mathbb{R}^2\rightarrow \mathbb{R}$ such that $f(x+iy)=u(x,y)+iv(x,y)$ is differentiable on $\mathbb{C}$, is what I've done below correct? I'm using the cauchy-riemann equations below. Is there any other (better) way?
$\frac{\partial u}{\partial x}=3x^2-3y^2-6xy+2x+2y=\frac{\partial v}{\partial y}\Rightarrow v=3x^2y-y^3-3xy^2+2xy+y^2+h(x)\\ -\frac{\partial u}{\partial y}=3x^2-3y^2+6xy-2x+2y=\frac{\partial v}{\partial y}\Rightarrow v=3x^2y+x^3-3xy^2+2xy-x^2+\bar h(y) \\ \Rightarrow v(x,y)=3x^2y-3xy^2+2xy+x^3-x^2+y^2-y^3 + c$