---
tag: 646PNPT1R6
kind: concept
name: Monogenic function
msc: [30G35]
sources:
  - url: https://en.wikipedia.org/wiki/Clifford_analysis
---

## Summary

A monogenic function is a function on an open subset of $\R^n$ with values in a [[Clifford algebra|A38B1QFK20]] that is annihilated by the [[Dirac operator|3DXGJTEQZH]].
It is the higher-dimensional analogue of a holomorphic function and the main object of [[Clifford analysis|DKVSAZQWF7]].
Monogenic functions satisfy a Cauchy integral formula, have local power series expansions, and their components are harmonic.

## Explanations

### L1 human

A holomorphic function solves the Cauchy–Riemann equations, which can be packed into one equation $\partial f / \partial \bar z = 0$.
In $\R^n$ one packs the derivatives into one operator $\sum_j e_j \partial_{x_j}$ with anticommuting units $e_j$.
A monogenic function is one sent to $0$ by this operator.
For example, a constant function is monogenic, and so is $x \mapsto x / \abs{x}^n$ away from $0$, where $x = \sum_j x_j e_j$; this is the Cauchy kernel up to a constant.