Bicomplex numbers as a normal complexified f-algebra

TL;DR

This paper explores the algebraic and topological properties of bicomplex numbers as a complexified f-algebra, introduces a D-trigonometric form, and generalizes key functions and formulas from complex analysis to the bicomplex setting.

Contribution

It develops a new geometric interpretation of bicomplex roots, establishes the equivalence of D-norm topologies, and generalizes the Riemann functional equation and Euler's reflection formula.

Findings

D-norms generate the same topology in B

Geometric interpretation of bicomplex roots as polyhedral tori

Generalization of Riemann functional equation and Euler's reflection formula

Abstract

The algebra B of bicomplex numbers is viewed as a complexification of the Archimedean f-algebra of hyperbolic numbers D. This lattice-theoretic approach allows us to establish new properties of the so-called D-norms. In particular, we show that D-norms generate the same topology in B. We develop the D-trigonometric form of a bicomplex number which leads us to a geometric interpretation of the nth roots of a bicomplex number in terms of polyhedral tori. We use the concepts developed, in particular that of Riesz subnorm of a D-norm, to study the uniform convergence of the bicomplex zeta and gamma functions. The main result of this paper is the generalization to the bicomplex case of the Riemann functional equation and Euler's reflection formula.