Geometry and Identity Theorems for Bicomplex Functions and Functions of a Hyperbolic Variable

摘要

Let D be the two-dimensional real algebra generated by 1 and by a hyperbolic unit k such that k2 = 1. This algebra is often referred to as the algebra of hyperbolic numbers. A function f : D. D is called D-holomorphic in a domain O. D if it admits derivative in the sense that limh. 0 f(z0+h)- f(z0) h exists for every point z0 in O, and when h is only allowed to be an invertible hyperbolic number. In this paper we prove that D-holomorphic functions satisfy an unexpected limited version of the identity theorem. We will offer two distinct proofs that shed some light on the geometry of D. Since hyperbolic numbers are naturally embedded in the four-dimensional algebra of bicomplex numbers, we use our approach to state and prove an identity theorem for the bicomplex case as well.