Cauchy–Riemann: satisfied at a point, holomorphic nowhere

Analytic FunctionsComplex Analysis, Algebra & Topologyfree

Why the equations alone are not enough, and the extra hypothesis that makes them work.

Step 1 / 4The equations

Writing $f = u + i v$ , complex differentiability at a point requires

$u_{x} = v_{y}, u_{y} = - v_{x} .$

These are necessary. They are not, on their own, sufficient.