Every time I search for a polar form of the Cauchy-Riemann equations, I find answers relating the derivatives of the real and complex parts of $f(z)$ to each other. That is: $\dfrac{\partial u}{\partial r} = \dfrac{1}r \dfrac{\partial v}{\partial \theta}$ and $\dfrac{1}r \dfrac{\partial u}{\partial \theta} = −\dfrac{\partial v}{\partial r}$.
This is emphatically not what I'm asking for. I mean the polar form of $f(z)$, that is:
$f(z)=r(z)e^{i\theta(z)}$
If you simply take the complex logarithm of both sides and take derivatives, it seems that:
$R_{90^{o}}[\frac{\nabla r(z)}{r(z)}] = \nabla \theta(z)$,
where $R_{90^{\circ}}$ is the rotation operator for $90$ degrees. Has anyone encountered this relation before? It's very useful for interpreting color-domain plots for example, if true. I just can't seem to find any name for this form of the C.R. equations.