---
tag: 1981Q0DGQW
kind: definition
name: Holomorphic map
msc: [32H02, 32A10]
defines: 3MQTWKBBBT
uses: [507C34W0ZT, 02BH3MCBQB, 047NDTEPCV]
sources:
  - url: https://en.wikipedia.org/wiki/Holomorphic_function
---

## Definition

Let $U \subseteq \C^n$ be an [[open set|02BH3MCBQB]] and let $F = (F_1, \dots, F_m) \colon U \to \C^m$ be a map with components $F_k \colon U \to \C$.
The map $F$ is *holomorphic* if each component $F_k$, $k = 1, \dots, m$, is a [[holomorphic function|507C34W0ZT]] on $U$.
If $V \subseteq \C^m$, a holomorphic map $U \to V$ is a holomorphic map $F \colon U \to \C^m$ with $F(U) \subseteq V$.

## Remarks

Equivalently, $F$ is differentiable at each point $a \in U$ in the real sense, and its derivative $DF(a) \colon \C^n \to \C^m$ is $\C$-linear.
This derivative is given by the complex [[Jacobian matrix|047NDTEPCV]] $\left( \frac{\partial F_k}{\partial z_j}(a) \right)_{k, j}$, an $m \times n$ matrix.
Example: $F(z_1, z_2) = (z_1 z_2, \ z_1 - z_2)$ is holomorphic from $\C^2$ to $\C^2$, since both components are polynomials.
Non-example: $G(z_1, z_2) = (z_1, \ \bar z_2)$ is not holomorphic, since its second component fails the Cauchy–Riemann equation $\partial G_2 / \partial \bar z_2 = 0$.
Compositions of holomorphic maps are holomorphic, by the chain rule.