Cauchy-Riemann equations (polar coordinates)

Suppose $A$ is an open set in $ℂ$ and $f(z)=f(r{e}^{i\theta })=u(r,\theta )+iv(r,\theta ):A\subset ℂ\to ℂ$ is a function. If the derivative of $f(z)$ exists at $z_0=(r_0,{\theta }_0)$. Then the functions $u$, $v$ at $z_0$ satisfy:

${\displaystyle \frac{\partial u}{\partial r}}$$=$${\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial v}{\partial \theta }}$

${\displaystyle \frac{\partial v}{\partial r}}$$=$$-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial u}{\partial \theta }}$

which are called Cauchy-Riemann equations in polar form.