Contents Introduction Chapter 1. Bicomplex and hyperbolic numbers 1.1. Bicomplex numbers 1.2. Conjugations and moduli. 1.3. The Euclidean norm on BC 1.4. Idempotent decompositions 1.5. A partial order on D and a hyperbolic-valued norm Chapter 2. Bicomplex functions and matrices 2.1. Bicomplex holomorphic functions 2.2. Bicomplex matrices Chapter 3. BC-modules 3.1. BC-modules and involutions on them 3.2. Constructing a a BC-module from two complex linear spaces. Chapter 4. Norms and inner products on BC-modules 4.1. Real-valued norms on bicomplex modules 4.2. D-valued norm on BC-modules 4.3. Bicomplex modules with inner product. 4.4. Inner products and cartesian decompositions 4.5. Inner products and idempotent decompositions. 4.6. Complex inner products on X induced by idempotent decompositions. 4.7. The bicomplex module BC n . 4.8. The ring H(C) of biquaternions as a BC-module Chapter 5. Linear functionals and linear operators on BCmodules 5.1. Bicomplex linear functionals 5.2. Polarization identities 5.3. Linear operators on BC-modules.

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. RESUMEN En este artículo introducimos elálgebra de números bicomplejos como una generalización del campo de números complejos. Describimos cómo definir funciones elementales en talesálgebras (polinomios y funciones exponenciales y trigonométricas) así como sus funciones inversas (raíces, logaritmos, funciones trigonométricas inversas). Nuestro objetivo es mostrar que una teoría de funciones sobre números bicomplejos es, en cierto sentido, una mejor generalización de la teoría de funciones holomorfas de una variable compleja, que la teoría de funciones holomorfas en dos variables complejas.

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