What is the partial derivative of the composition of two complex functions?

Question

I'm trying to prove that if we have an analytic function $f : U \rightarrow \mathbb{C}$ and a harmonic function $w : V \rightarrow \mathbb{R}$ (where of course $f(U)\subset V$) then its composition, namely $w \circ f$, is also harmonic.

I know that the hypothesis implies that if $f= u +iv$ then:

$$\frac{\partial ^2 u}{\partial x^2}+ \frac{\partial ^2 u}{\partial y^2} = 0$$

and

$$\frac{\partial ^2 v}{\partial x^2}+ \frac{\partial ^2 v}{\partial y^2} = 0$$

For $w$ we know:

$$\frac{\partial ^2 w}{\partial x^2}+ \frac{\partial ^2 w}{\partial y^2} = 0$$

It seems that I should try to calculate $\displaystyle\frac{\partial ^2 (w \circ f)}{\partial x^2}$ and $\displaystyle\frac{\partial ^2 (w \circ f)}{\partial y^2}$, to factorize any of the terms above, but I still don't quite understand how to calculate $\displaystyle\frac{\partial (w \circ f)}{\partial x}$ even (which is where the title of the question comes from). I'm sorry if this last part is obvious, but I'm just starting to work with the analysis of complex functions. Any help would be apricated, thank you.

Answer 1

Being harmonic is a local property. Locally, $w=\operatorname{Re}g$ for some analytice function $g$. Therefore, $w\circ f=\operatorname{Re}(g)\circ f=\operatorname{Re}(g\circ f)$, and therefore it is harmonic too.

Answer 2

The function $f$ is analytic so $\dfrac{\partial f}{\partial \overline{z}}=0$ and the function $w$ is harmonic then $\dfrac{\partial}{\partial \overline{z}}\dfrac{\partial}{\partial z}w=0$, The composition $F=w \circ f$ is \begin{align} \dfrac{\partial F}{\partial z} &= \dfrac{\partial w \circ f}{\partial f}\dfrac{\partial f}{\partial z}+\dfrac{\partial w \circ f}{\partial \overline{f}}\dfrac{\partial \overline{f}}{\partial z} \\ &= \dfrac{\partial w \circ f}{\partial f}\dfrac{\partial f}{\partial z}\\ \dfrac{\partial}{\partial \overline{z}}\dfrac{\partial F}{\partial z} &= \dfrac{\partial}{\partial \overline{z}} \dfrac{\partial w \circ f}{\partial f}.\dfrac{\partial f}{\partial z} + \dfrac{\partial}{\partial \overline{z}} \dfrac{\partial f}{\partial z}.\dfrac{\partial w \circ f}{\partial f} \\ &= 0.\dfrac{\partial f}{\partial z} + 0.\dfrac{\partial w \circ f}{\partial f} \\ &= 0 \end{align}