On Decompositions of H-holomorphic functions into quaternionic power series

TL;DR

This paper explores the parallels between quaternionic and complex holomorphic functions, establishing criteria for quaternionic power series convergence and demonstrating their similarity to complex cases.

Contribution

It introduces a unified notion of holomorphic functions applicable to both quaternionic and complex contexts, with convergence criteria and power series expansion properties.

Findings

Quaternionic power series converge similarly to complex ones.

Every quaternionic holomorphic function is H-analytic.

Power series expansions have identical convergence radii in both settings.

Abstract

Based on the full similarity in algebraic properties and differentiation rules between quaternionic (H-) holomorphic and complex (C-) holomorphic functions, we assume that there exists one holistic notion of a holomorphic function that has a H-representation in the case of quaternions and a C-representation in the case of complex variables. We get the essential definitions and criteria for a quaternionic power series convergence, adapting complex analogues to the quaternion case. It is established that the power series expansions of any holomorphic function in C- and H-representations are similar and converge with identical convergence radiuses. We define a H-analytic function and prove that every H-holomorphic function is H-analytic. Some examples of power series expansions are given.