Cauchy–Riemann operator

Summary

The Cauchy–Riemann operator is the first-order differential operator $\partial / \partial \bar z = \tfrac12 (\partial_x + i \partial_y)$ on complex-valued functions of $z = x + iy$. A continuously differentiable function $f$ is holomorphic exactly when $\partial f / \partial \bar z = 0$, which is a compact form of the Cauchy–Riemann equations. It is the simplest elliptic operator, with fundamental solution $1/(\pi z)$. Its several-variable version, the $\bar\partial$ operator, is a main tool of complex analysis and complex geometry.