A First Course in Complex Analysis · Problem 25

Examples of Functions — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · cauchy-riemann-equations

Problem

Prove that the Cauchy--Riemann equations take on the following form in polar coordinates:

\frac{\partial u}{\partial r} = \frac{1}{r}\frac{\partial v}{\partial \phi} \quad \text{and} \quad \frac{1}{r}\frac{\partial u}{\partial \phi} = -\frac{\partial v}{\partial r}.