---
tag: 3FE2AQS534
kind: definition
name: Quaternionic analysis
msc: [30G35]
defines: SAT7ST5WP7
uses: [0GE4DVQKR8, 0EA5BTA0CX, 09EGC6F8PT, 0E0B6RA8K0]
sources:
  - url: https://en.wikipedia.org/wiki/Quaternionic_analysis
---

## Definition

Let $\mathbb{H}$ be the [[quaternions|0GE4DVQKR8]], the real [[division ring|0EA5BTA0CX]] with basis $1, i, j, k$ and $i^2 = j^2 = k^2 = ijk = -1$.
Write $q = x_0 + x_1 i + x_2 j + x_3 k$ with $x_0, \ldots, x_3 \in \R$, and identify $\mathbb{H}$ with $\R^4$.
Let $U \subset \mathbb{H}$ be open.
A function $f \colon U \to \mathbb{H}$ with continuous first [[partial derivatives|09EGC6F8PT]] is *left regular*, or *Fueter regular*, if
$$\frac{\partial f}{\partial x_0} + i \frac{\partial f}{\partial x_1} + j \frac{\partial f}{\partial x_2} + k \frac{\partial f}{\partial x_3} = 0 \quad \text{on } U.$$
*Quaternionic analysis* is the study of functions $U \to \mathbb{H}$ by analogy with complex analysis, and above all of regular functions.

## Remarks

Example: $f(q) = x_1 - i x_0$ is left regular, since $\partial_{x_0} f + i \partial_{x_1} f = -i + i = 0$ and the other two derivatives vanish.

Non-example: $f(q) = q$ is not left regular.
The sum is $1 + i \cdot i + j \cdot j + k \cdot k = 1 - 3 = -2 \ne 0$.
So, unlike in complex analysis, the identity function and polynomials in $q$ are in general not regular.

Write $\bar\partial = \partial_{x_0} + i \partial_{x_1} + j \partial_{x_2} + k \partial_{x_3}$ and $\partial = \partial_{x_0} - i \partial_{x_1} - j \partial_{x_2} - k \partial_{x_3}$.
Then $\partial \bar\partial$ is the [[Laplacian|0E0B6RA8K0]] of $\R^4$, so each component of a regular function of class $C^2$ is harmonic.
Right regular functions are defined with the units multiplied on the right of the derivatives.