Calculus with a Quaternionic Variable

TL;DR

This paper develops a calculus framework for quaternionic variables, overcoming non-commutativity obstacles, and introduces a new first-order expansion formula that advances quaternionic differential and integral calculus.

Contribution

It presents a novel first-order expansion formula for quaternionic functions, addressing non-commutativity issues in quaternionic calculus.

Findings

Derived a compact formula for the first-order term involving F' and quaternion conjugates.

Progressed in quaternionic integration methods.

Overcame key obstacles posed by non-commutativity in quaternion calculus.

Abstract

Most of theoretical physics is based on the mathematics of functions of a real or a complex variable; yet we frequently are drawn to try extending our reach to include quaternions. The non-commutativity of the quaternion algebra poses obstacles for the usual manipulations of calculus; but we show in this paper how many of those obstacles can be overcome. The surprising result is that the first order term in the expansion of F(x+delta) is a compact formula involving both F'(x) and [F(x) - F(x*)]/(x-x*). This advance in the differential calculus for quaternionic variables also leads us to some progress in studying integration.