Bicomplex Numbers and their Elementary Functions

M. E. Luna; M. Shapiro; D. C. Struppa; A. Vajiac

doi:10.4067/S0719-06462012000200004

Authors

M. E. Luna Departamento de Matemáticas, E.S.F.M del I.P.N., México.
M. Shapiro Departamento de Matemáticas, E.S.F.M del I.P.N., México.
D. C. Struppa Schmid College of Science and Technology, Chapman University, Orange California, USA.
A. Vajiac Schmid College of Science and Technology, Chapman University, Orange California, USA.

DOI:

https://doi.org/10.4067/S0719-06462012000200004

Keywords:

Bicomplex numbers, Elementary functions

Abstract

In this paper we introduce the algebra of bicomplex numbers as a generalization of the field of complex numbers. We describe how to define elementary functions in such an algebra (polynomials, exponential functions, and trigonometric functions) as well as their inverse functions (roots, logarithms, inverse trigonometric functions). Our goal is to show that a function theory on bicomplex numbers is, in some sense, a better generalization of the theory of holomorphic functions of one variable, than the classical theory of holomorphic functions in two complex variables.