---
tag: DKVSAZQWF7
kind: concept
name: Clifford analysis
msc: [30G35, 15A66]
sources:
  - url: https://en.wikipedia.org/wiki/Clifford_analysis
same_as:
  wikidata: Q5133083
  wikipedia: Clifford analysis
---

## Summary

Clifford analysis extends complex analysis to $\R^n$ by letting functions take values in a [[Clifford algebra|A38B1QFK20]].
Its central objects are the [[monogenic functions|646PNPT1R6]], the functions annihilated by a [[Dirac operator|3DXGJTEQZH]].
The square of the Dirac operator is the Laplacian up to sign.
So the theory refines harmonic analysis, as holomorphic functions refine harmonic functions in the plane.
It has a Cauchy integral formula, Taylor and Laurent series, and links to singular integrals, spin geometry and representation theory.
It grew out of [[quaternionic analysis|SAT7ST5WP7]], which is closely related to its low-dimensional cases.

## Explanations

### L1 human

In the plane, holomorphic functions are the solutions of the Cauchy–Riemann equations, a first-order system whose "square" is the Laplace equation.
In higher dimensions there is no such system with complex numbers alone.
A Clifford algebra supplies anticommuting units $e_1, \ldots, e_n$ with $e_j^2 = -1$, and with them the operator $\sum_j e_j \partial_{x_j}$ squares to minus the Laplacian.
Functions killed by this operator behave much like holomorphic functions: they have an integral formula, power series, and a maximum principle.