I had the following shower-thought this morning:
Let $f: \Bbb R^2 \to \Bbb R^2$ be given by $$ f(x,y) = (u(x,y),v(x,y)), $$ and let $g:\Bbb C \to \Bbb C$ be the corresponding function satisfying $$ g(x+iy) = u(x,y) + iv(x,y). $$ Let $U \subset \Bbb C$ be open (for convenience, we identify it with the corresponding Then the following conditions are equivalent:
- $g$ is analytic over $U$
- For every $(x,y) \in U$, $f$ satisfies the Cauchy-Riemann equations
- For every $(x,y) \in U$, there exists some $a + bi \in C$ such that the Jacobian map $Df(x,y):\Bbb R^2 \to \Bbb R^2$ defined by $$ Df(x,y)(\hat x, \hat y) = \pmatrix{\dfrac{\partial u}{\partial x} & \dfrac{\partial u}{\partial y} \\[3 pt] \dfrac{\partial v}{\partial x} & \dfrac{\partial v}{\partial y}}\pmatrix{\hat x \\ \hat y} $$ is just multiplication by the complex number $(a+bi)$.
Proof: $2 \iff 3$: set $a = \frac{\partial u}{\partial x}$ and $b = \frac{\partial v}{\partial x}$.
My question is as follows: is this a useful or interesting perspective? Is it common in some application of complex analysis?
I found this section of a wiki article, which is related but more specific than my point. If you could point me in the direction of a reference, that would be appreciated.
Thanks for reading.