---
tag: Z8NJ4KHGAT
kind: definition
name: Monogenic function
msc: [30G35]
defines: 646PNPT1R6
uses: [A38B1QFK20, 09EGC6F8PT, 0E0B6RA8K0]
sources:
  - url: https://en.wikipedia.org/wiki/Clifford_analysis
---

## Definition

Let $n \ge 2$ and let $\mathbb{R}_{0,n}$ be the real [[Clifford algebra|A38B1QFK20]] generated by $e_1, \ldots, e_n$ with $e_j^2 = -1$ and $e_j e_k = -e_k e_j$ for $j \ne k$.
Let $D = \sum_{j=1}^{n} e_j \dfrac{\partial}{\partial x_j}$ be the [[Dirac operator|3DXGJTEQZH]] on $\R^n$.
Let $U \subset \R^n$ be open and let $f \colon U \to \mathbb{R}_{0,n}$ have continuous first [[partial derivatives|09EGC6F8PT]].
The function $f$ is *left monogenic* on $U$ if
$$D f = \sum_{j=1}^{n} e_j \frac{\partial f}{\partial x_j} = 0 \quad \text{on } U.$$
It is *right monogenic* if $\sum_{j} \dfrac{\partial f}{\partial x_j} e_j = 0$ on $U$.

## Remarks

Example: identify $x \in \R^n$ with $\sum_j x_j e_j$, so that $x^2 = -\abs{x}^2$.
The Cauchy kernel $E(x) = \dfrac{\bar x}{\abs{x}^n}$ with $\bar x = -x$ is left and right monogenic on $\R^n \setminus \set{0}$.
A simpler example is $f(x) = x_1 e_1 - x_2 e_2$, since $Df = e_1 e_1 - e_2 e_2 = -1 + 1 = 0$.

Non-example: $f(x) = x = \sum_j x_j e_j$ is not monogenic, since $Df = \sum_j e_j^2 = -n \ne 0$.

Since $D^2 = -\Delta$, with $\Delta$ the [[Laplacian|0E0B6RA8K0]], each real component of a left monogenic function of class $C^2$ is harmonic.
Many authors use instead the generalized Cauchy–Riemann operator $\partial_{x_0} + \sum_{j=1}^{n} e_j \partial_{x_j}$ on $\R^{n+1}$; for $n = 1$ it gives back holomorphic functions.