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 $$$$ is the gradient of a harmonic function $$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.
• You're right that $v_1$ and $v_2$ can each be parts (real or imaginary) of holomorphic functions. But you haven't taken into account the relations between the partial derivatives of $v_1$ and $v_2$. They tell you that $v_2 + i v_1$ is holomorphic. – Matthew Leingang Jul 18 at 19:17