Bicomplex algebra and function theory

TL;DR

This paper explores the theory of holomorphic functions on bicomplex numbers, establishing foundational properties, integral theorems, and connections to Laplace equations, extending quaternionic function theory.

Contribution

It develops the theory of bicomplex holomorphic functions, including derivatives, integrals, and their relation to Laplace equations, introducing new classes of functions.

Findings

Holomorphic bicomplex functions satisfy complexified Cauchy-Riemann equations.

Bicomplex integrals are path-independent under certain conditions.

Connections between bicomplex functions and four-dimensional Laplace equations are established.

Abstract

This treatise investigates holomorphic functions defined on the space of bicomplex numbers introduced by Segre. The theory of these functions is associated with Fueter's theory of regular, quaternionic functions. The algebras of quaternions and bicomplex numbers are developed by making use of so-called complex pairs. Special attention is paid to singular bicomplex numbers that lack an inverse. The elementary bicomplex functions are defined and their properties studied. The derivative of a bicomplex function is defined as the limit of a fraction with nonsingular denominator. The existence of the derivative amounts to the validity of the complexified Cauchy-Riemann equations, which characterize the holomorphic bicomplex functions. It is proved that such a function has derivatives of all orders. The bicomplex integral is defined as a line integral. The condition for path independence and the bicomplex generalizations of Cauchy's theorem and integral formula are given. Finally, the relationship between the bicomplex functions and different forms of the Laplace equation is considered. In particular, the four-dimensional Laplace equation is factorized using quaternionic differential operators. The outcome is new classes of bicomplex functions including Fueter's regular functions. It is shown that each class contains differentiable functions.