On a three dimensional analogue to the holomorphic z-powers

TL;DR

This paper extends holomorphic power and Laurent series concepts from complex analysis to three dimensions using hypercomplex function theory, introducing new orthogonal series and basis representations in quaternionic spaces.

Contribution

It provides a constructive generalization of power and Laurent series in 3D quaternionic spaces, including explicit basis representations and a generalized Laurent expansion for spherical shells.

Findings

Developed orthogonal Fourier and Taylor series in quaternionic L2 space.

Established explicit recurrence and closed-form formulas for basis elements.

Defined a generalized Laurent series expansion for spherical shells.

Abstract

The main objective of this article is a constructive generalization of the holomorphic power and Laurent series expansions in C to dimension 3 using the framework of hypercomplex function theory. For this reason, deals the first part of this article with generalized Fourier & Taylor series expansions in the space of square integrable quaternion-valued functions which possess peculiar properties regarding the hypercomplex derivative and primitive. In analogy to the complex one-dimensional case, both series expansions are orthogonal series with respect to the unit ball in R^3 and their series coefficients can be explicitly (one-to-one) linked with each other. Furthermore, very compact and efficient representation formulae (recurrence, closed-form) for the elements of the orthogonal bases are presented. The latter results are then used to construct a new orthonormal bases of outer solid spherical monogenics in the space of square integrable quaternion-valued functions. This finally leads to the definition of a generalized Laurent series expansion for the spherical shell.