Chain rule for functions on a complex domain

Question

Im am confronted with the following claim, which I simply cannot seem to prove:

Let $s:D\setminus \{0\}\subset \mathbb{C}\to \mathbb{R}^n$, let $w(z)=\left(|s_x|^2-|s_y|^2+2i\langle s_x,s_y\rangle \right)$ where $\langle, \rangle$ is the euclidean inner product on $\mathbb{R}^n$. Then it holds that $$Re(w(z)z^2)=|s_\theta(z)|^2-|z|^2|s_r(z)|^2.$$

where it is implicitly assumed that $r,\theta$ are the usual polar and $|\cdot|$ is here used simultaneously for the complex module and for the norm.

I've tried the following:

\begin{align*}\Re(w(z)z^2) & =\Re\left(\left(|s_x|^2-|s_y|^2+2i\langle s_x,s_y\rangle \right)(x^2-y^2+2ixy)\right) \\ & =\left(|s_x|^2-|s_y|^2\right)(x^2-y^2)-4\langle s_x,s_y\rangle xy. \end{align*}

On the other hand, using the chain rule, we have that: \begin{align*} s_\theta&=s_xx_\theta+s_yy_\theta=s_xy+s_yx\\ s_r& =s_xx_r+s_yy_r=\frac{s_x x+s_y y}{|z|} \end{align*} Consequently \begin{align*}|s_\theta|^2-|z|^2|s_r|^2& =|s_x|^2y^2-2\langle s_x,s_y \rangle xy+ |s_y|^2x^2-|s_x|^2x^2-2\langle s_x,s_y \rangle xy- |s_y|^2y^2 \\ & =|s_x|^2(y^2-x^2)+|s_y|^2(-y^2+x^2)-4\langle s_x,s_y \rangle xy\\ & =(|s_x|^2-|s_y|^2)(y^2-x^2)-4\langle s_x,s_y \rangle xy. \end{align*}

But confronting the two sides, there appears to be a mistake in my calculation. I am aware that I did the calculation as I had been in $\mathbb{R}^2$, but taking into account the "complex" variable made the the result inconsistent.

Can someone help me out with this issue?

Many thanks in advance

Answer 1

I think OPs calculation is correct and the claim is wrong. Let's make a plausibility check with a simple example:

We set \begin{align*} &s:D\setminus \{0\}\subset \mathbb{C}\to \mathbb{R}\\ &s(z)=s(x+iy)=x \end{align*}

and we obtain \begin{align*} w(z)&=\left(|s_x|^2-|s_y|^2+2i\langle s_x,s_y \rangle\right)\\ &=\left(1-0+2i\langle 1,0\rangle\right)\\ &=1 \end{align*} It follows \begin{align*} \Re(w(z)z^2)=\Re(z^2)=x^2-y^2\tag{1} \end{align*}

on the other hand, since \begin{align*} x&=x(r,\theta)=r\cdot\cos \theta\\ y&=y(r,\theta)=r\cdot\sin \theta \end{align*} with \begin{align*} s(z)&=x=r\\ \end{align*} we get \begin{align*} s_\theta&=s_xx_\theta+s_yy_\theta\\ &=s_x\cdot 0+0\\ &=0\\ s_r&=s_xx_r+s_yy_r\\ &=s_x\\ &=1 \end{align*}

We obtain \begin{align*} \left|s_\theta\right|^2-\left| z\right|^2\left|s_r\right|^2 =-|z|^2=-x^2+y^2\tag{2} \end{align*}

with (2) differing from (1) by a factor $-1$ in accordance with OPs calculation.