Wirtinger derivatives

Summary

The Wirtinger derivatives $\partial / \partial z$ and $\partial / \partial \bar z$ are two first-order differential operators on functions of $z = x + iy$, see the definition. They let one treat $z$ and $\bar z$ as if they were independent variables. A continuously differentiable function is holomorphic exactly when its $\partial / \partial \bar z$ derivative is zero, which is a compact form of the Cauchy–Riemann equations.