Bicomplex numbers as a normal complexified f-algebra

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.

On Hyperbolic Integers

“…As an application to fractal geometry, a concept of Cantor sets in hyperbolic numbers was developed by Balankin et al [4] and Téllez-Sánchez et al [24]. Recently, the authors of the present paper used in [8] lattice-theoretical results to go further in the development of the theory of bicomplex zeta function. Further applications are found in [13,14,17,18,20].…”