We know that, by applying a coordinate change, we have the polar form of CR-equation which is given by: $$ \begin{split} \frac{\partial u}{\partial r} &= \frac{1}{r}\frac{\partial v}{\partial \theta}, \\ \frac{1}{r}\frac{\partial u}{\partial \theta} &= - \frac{\partial v}{\partial r}. \end{split} $$
This equation can be used to check whether a function is holomorphic at some point, correct? I find there is no reference makes the theorem in a formal statement, and I found a that this equation can not check whether $f(z)$ is holomorphic at $z = \{z| z\in \Bbb{R}_{-}\}$ since over this axis the re-parametrization $u(r,\theta )$ is not continuous, correct? Is there some formal criterion for holomorphic function in polar coordinate?