Question: Let $(v_1, v_2)$ be a pair of harmonic functions on a disc $U \subseteq \mathbb{C}$. Suppose that
$\frac{\partial v_1 }{\partial y} = \frac{\partial v_2}{\partial x}$ and $\frac{\partial v_1}{\partial x} + \frac{\partial v_2}{\partial y} = 0$. Show that $<v_1, v_2> $ is the gradient of a harmonic function $h$.
Note that $v_1, v_2$ are real valued harmonic functions. Now I thought of it this way. Since both $v_1, v_2$ are harmonic, there are holomorphic functions $F_1, F_2$ such that $Re(F_1) = v_1$ and $Re(F_2) = v_2$. Since $F_1$ and $F_2$ are holomorphic, they have holomorphic anti-derivatives. Thus, there are holomorphic functions, $H_1,H_2$, so that $\frac{\partial H_1 }{\partial z} = F_1$ and $\frac{\partial H_2 }{\partial z} = F_2$. I am not sure where to proceed at this point. I tried coming up a function $H$ which involves $v1,v2$ but the problem is harmonic functions does not imply holomorphic functions.
Do you guys have any suggestions? Thank you so much.