Math 2.0

Functions of a complex variable Definition

Quaternionic analysis

Builds on The quaternionsDivision ringPartial derivativeLaplacian

Definition

Let ℍ\mathbb{H} be the quaternions, the real division ring with basis 1,i,j,k1, i, j, k and i2=j2=k2=ijk=−1i^2 = j^2 = k^2 = ijk = -1. Write q=x0+x1i+x2j+x3kq = x_0 + x_1 i + x_2 j + x_3 k with x0,…,x3∈ℝx_0, \ldots, x_3 \in \R, and identify ℍ\mathbb{H} with ℝ4\R^4. Let U⊂ℍU \subset \mathbb{H} be open. A function f ⁣:U→ℍf \colon U \to \mathbb{H} with continuous first partial derivatives is left regular, or Fueter regular, if ∂f∂x0+i∂f∂x1+j∂f∂x2+k∂f∂x3=0on U.\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→ℍU \to \mathbb{H} by analogy with complex analysis, and above all of regular functions.

Remarks

Example: f(q)=x1−ix0f(q) = x_1 - i x_0 is left regular, since ∂x0f+i∂x1f=−i+i=0\partial_{x_0} f + i \partial_{x_1} f = -i + i = 0 and the other two derivatives vanish.

Non-example: f(q)=qf(q) = q is not left regular. The sum is 1+i⋅i+j⋅j+k⋅k=1−3=−2≠01 + 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 qq are in general not regular.

Write ∂ˉ=∂x0+i∂x1+j∂x2+k∂x3\bar\partial = \partial_{x_0} + i \partial_{x_1} + j \partial_{x_2} + k \partial_{x_3} and ∂=∂x0−i∂x1−j∂x2−k∂x3\partial = \partial_{x_0} - i \partial_{x_1} - j \partial_{x_2} - k \partial_{x_3}. Then ∂∂ˉ\partial \bar\partial is the Laplacian of ℝ4\R^4, so each component of a regular function of class C2C^2 is harmonic. Right regular functions are defined with the units multiplied on the right of the derivatives.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Division ring, concept, no evidence levelDivision ringconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptGroup, concept, no evidence levelGroupconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptRing, definition, no evidence levelRingdefinitionDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptUnit of a ring, definition, no evidence levelUnit of a ringdefinitionGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptDivision ring, definition, no evidence levelDivision ringdefinitionLaplacian, concept, no evidence levelLaplacianconceptThe quaternions, example, no evidence levelThe quaternionsexampleQuaternionic analysis, definition, no evidence levelQuaternionic analysisdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Division ring concept
  2. Euclidean space object
  3. Function concept
  4. Group concept
  5. Total derivative concept
  6. Function of several variables concept
  7. Vector-valued function concept
  8. Ring definition
  9. Differentiable function of several variables concept
  10. Partial derivative concept
  11. Vector field concept
  12. Unit of a ring definition
  13. Gradient concept
  14. Divergence concept
  15. Division ring definition
  16. Laplacian concept
  17. The quaternions example
  18. Quaternionic analysis definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Quaternionic_analysis
About this page
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URIm2://3FE2AQS534
Kinddefinition
WritingW0 stub
MSC30G35
DefinesQuaternionic analysis
UsesThe quaternions, Division ring, Partial derivative, Laplacian
Changed2 November 2026

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