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.
In this paper we study the singularities of holomorphic functions of bicomplex variables introduced by G. B. Price (An Introduction to Multicomplex Spaces and Functions, Dekker, New York, 1991). In particular, we use computational algebra techniques to show that even in the case of one bicomplex variable, there cannot be compact singularities. The same techniques allow us to prove a duality theorem for such functions.
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