---
tag: 507C34W0ZT
kind: concept
name: Holomorphic function of several variables
msc: [32A10]
specializes: [BH57D2FBNR]
sources:
  - url: https://en.wikipedia.org/wiki/Function_of_several_complex_variables
---

## Summary

A holomorphic function of several variables is a function on an open subset of $\C^n$ that is complex differentiable at every point.
It extends the notion of a [[holomorphic function|0NAHXR1FRH]] of one variable.
Such a function is holomorphic in each variable separately and infinitely differentiable.
It is locally the sum of a convergent [[power series in several variables|R3QTCPTJK7]].
These functions are the main objects of the theory of [[several complex variables|BH57D2FBNR]].

## Explanations

### L1 human

A function of one complex variable is holomorphic when it has a complex derivative.
For several variables, one asks that near each point the function is well approximated by a complex linear function.
Polynomials in $z_1, \dots, z_n$, such as $z_1^3 - 2 z_1 z_2$, are holomorphic on all of $\C^n$.
The function $1/(1 - z_1 - z_2)$ is holomorphic where $z_1 + z_2 \ne 1$.
A function that involves $\bar z_1$, such as $\abs{z_1}^2$, is not holomorphic.