Clifford analysis

Summary

Clifford analysis extends complex analysis to $\R^n$ by letting functions take values in a Clifford algebra. Its central objects are the monogenic functions, the functions annihilated by a Dirac operator. 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, which is closely related to its low-dimensional cases.