Let \(\Omega\subseteq\CC\sm\ts{0}\) be open and let \(f=u+iv\) be holomorphic on \(\Omega\), with \(u,v\) real-valued. Writing \(u\) and \(v\) as functions of \((r,\theta)\) through \(z=re^{i\theta}\), \(r>0\), \[ \frac{\partial u}{\partial r}=\frac{1}{r}\frac{\partial v}{\partial\theta}\qquad\text{and}\qquad\frac{1}{r}\frac{\partial u}{\partial\theta}=-\frac{\partial v}{\partial r}. \]
With \(x=r\cos\theta\) and \(y=r\sin\theta\), the chain rule gives \[ \begin{aligned} u_r&=u_x\cos\theta+u_y\sin\theta, & u_\theta&=r(-u_x\sin\theta+u_y\cos\theta),\\ v_r&=v_x\cos\theta+v_y\sin\theta, & v_\theta&=r(-v_x\sin\theta+v_y\cos\theta). \end{aligned} \] Substituting the Cauchy–Riemann equations \(v_x=-u_y\) and \(v_y=u_x\) gives \[ v_\theta=r(u_y\sin\theta+u_x\cos\theta)=ru_r,\qquad v_r=-u_y\cos\theta+u_x\sin\theta=-\frac1r u_\theta. \]