---
tag: 698R12MY4G
kind: definition
name: Wirtinger derivatives
msc: [30A99, 32A10]
defines: WMXDVF36ZT
uses: [05AXG3QRKR, 0DG0Z72CQD, 0T71SSYZ39]
sources:
  - url: https://en.wikipedia.org/wiki/Wirtinger_derivatives
---

## Definition

Let $U \subseteq \C$ be an open set and write $z = x + iy$ with $x, y$ real.
Let $f \colon U \to \C$ be a function whose [[partial derivatives|05AXG3QRKR]] $\partial f / \partial x$ and $\partial f / \partial y$ exist at a point $z \in U$.
The *Wirtinger derivatives* of $f$ at $z$ are
$$\frac{\partial f}{\partial z} = \frac{1}{2} \left( \frac{\partial f}{\partial x} - i \frac{\partial f}{\partial y} \right), \qquad \frac{\partial f}{\partial \bar z} = \frac{1}{2} \left( \frac{\partial f}{\partial x} + i \frac{\partial f}{\partial y} \right).$$

## Remarks

Example: for $f(z) = \abs{z}^2 = x^2 + y^2$, one gets $\partial f / \partial z = \frac{1}{2}(2x - 2iy) = \bar z$ and $\partial f / \partial \bar z = \frac{1}{2}(2x + 2iy) = z$.
Non-example of holomorphy: since $\partial f / \partial \bar z = z \neq 0$ for $z \neq 0$, the function $\abs{z}^2$ is not [[holomorphic|0DG0Z72CQD]] on any open set.
For $f$ of class $C^1$ on $U$, the equation $\partial f / \partial \bar z = 0$ on $U$ is equivalent to the [[Cauchy–Riemann equations|0T71SSYZ39]], so it holds exactly when $f$ is holomorphic on $U$; then $\partial f / \partial z = f'$.
The operators obey the rules $\partial z / \partial z = 1$, $\partial \bar z / \partial z = 0$, $\partial z / \partial \bar z = 0$ and $\partial \bar z / \partial \bar z = 1$, together with linearity and the product rule.
The Laplacian is $\Delta f = 4 \, \dfrac{\partial^2 f}{\partial z \, \partial \bar z}$ for $f$ of class $C^2$.
The notations $\partial_z f$ and $\partial_{\bar z} f$ are also common.