Cauchy-Riemann equations in arbitrary coordinates? The CR equations in rectangular coordinates can be written as one equation in the following way:
$$\frac{\partial f}{\partial x} = \frac{1}{i} \frac{\partial f}{\partial y}$$
Likewise, in polar coordinates, it's:
$$\frac{\partial f}{\partial r} = \frac{1}{ir} \frac{\partial f}{\partial \theta}.$$
Now the factor of $1/r$ that appears looks curiously like the Jacobian between the two coordinate systems, so I was wondering whether for an arbitrary coordinate choice, $p(x,y), q(x,y)$, is it true that
$$\frac{\partial f}{\partial p} = \frac{1}{i J} \frac{\partial f}{\partial q}$$
where $$J= \left| \frac{D[x,y]}{D[p,q]} \right| \ \ ???$$
I was unfortunately unable to find any mention of this on the internet, meaning it's probably untrue. If so, is there a similarly straightforward way of writing down the equations in arbitrary coordinate systems?
 A: It's not quite as easy. When you have arbitrary coordinates $(\xi,\eta)$, the change of coordinates for the differential operators is given by
$$\begin{pmatrix}\frac{\partial}{\partial x}\\\frac{\partial}{\partial y}\end{pmatrix} = \begin{pmatrix}\frac{\partial \xi}{\partial x} & \frac{\partial \eta}{\partial x}\\ \frac{\partial \xi}{\partial y} & \frac{\partial \eta}{\partial y} \end{pmatrix} \cdot \begin{pmatrix}\frac{\partial}{\partial \xi}\\\frac{\partial}{\partial \eta}\end{pmatrix}.$$
The Cauchy Riemann equations say that $f$ is annihilated by the differential operator
$$\frac{\partial}{\partial x} + i \frac{\partial}{\partial y} = \left(\frac{\partial \xi}{\partial x} + i\frac{\partial\xi}{\partial y}\right) \frac{\partial}{\partial \xi} + \left(\frac{\partial \eta}{\partial x} + i\frac{\partial\eta}{\partial y}\right) \frac{\partial}{\partial \eta}.$$
You get something as nice as $\frac{\partial}{\partial r} + \frac{i}{r}\frac{\partial}{\partial \theta}$ only for particularly nice coordinate systems. The niceness of the polar coordinates is that $\left(\frac{\partial}{\partial r},\, \frac{\partial}{\partial \theta} \right)$ is (outside the origin, of course) a positively oriented orthogonal basis of the tangent space of $\mathbb{C}$, and the ratio of the lengths depends only on the absolute modulus of the point.
