Past exam of the short options course of the University of Oxford / 2021 / 4 / c / i / Solution

Write $a={\partial }_1{f}_1={\partial }_2{f}_2$ and $b={\partial }_2{f}_1=-{\partial }_1{f}_2$. The Cauchy-Riemann equations giveConsequentlyEquivalently, it is singular exactly when all four partial derivatives vanish. The sum of squares is over $\mathbb{R}$, so no cancellation is possible.