Suppose $A$ is an open set in $\mathbb{C}$ and $f(z)=f(re^{i\theta})=u(r,\theta)+iv(r,\theta): A\subset\mathbb{C} \to \mathbb{C}$ 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: \begin{eqnarray*} \frac{\partial u}{\partial r} & = & \frac{1}{r}\frac{\partial v}{\partial \theta}\\ \frac{\partial v}{\partial r} & = & -\frac{1}{r}\frac{\partial u}{\partial \theta} \end{eqnarray*}which are called Cauchy-Riemann equations in polar form.