Complex analyticity and vector potentials?

Question

In this answer, we have the expression:

$$\begin{align}&\frac{\partial \phi}{\partial x} = \phantom{-}\frac{\partial \bf A_z} {\partial y}\\& \frac{\partial \phi}{\partial y} = -\frac{\partial {\bf A_z}} {\partial x}\end{align}$$

In complex analysis we have the Cauchy-Riemann equations, which are necessary (and sufficient?) for a complex valued function $$a+bi \to f(a,b) = u(a,b) + iv(a,b), \hspace{1cm} a,b,u,v\in \mathbb R$$ to be complex analytic:

$$\begin{align}&\frac{\partial u}{\partial x} = \phantom{-}\frac{\partial v}{\partial y}\\&\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\end{align}$$

We can identify $v = \bf A_z$, $u = \phi$

Is this a just a coincidental curiosity or something useful?

Answer 1

It is not a coincidence. The same linear operator that rotates a vector field by 90 degrees (e.g., $(P, Q)$ becomes $(Q, -P)$) appears in both contexts. It is a 2-dimensional case of the Hodge star operator. You will find both the Cauchy-Riemann equation and the curl operator mentioned in that article.

The Cauchy-Riemann equations say that $\nabla v$ is $\nabla u$ rotated by 90 degrees. This reflects the concept of conformality: the map $u+iv$ transforms the original orthogonal system of coordinates to another orthogonal system of coordinates (infinitesimally).