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

Abstract

Let $$\mathbb{D}$$ be the two-dimensional real algebra generated by 1 and by a hyperbolic unit k such that $$k^{2} = 1$$. This algebra is often referred to as the algebra of hyperbolic numbers. A function $$f : \mathbb{D} \rightarrow \mathbb{D}$$ is called $$\mathbb{D}$$-holomorphic in a domain $$\Omega \subset \mathbb{D}$$ if it admits derivative in the sense that $${\rm lim}_{h\rightarrow{0}}\frac{f({\mathfrak{z}_{0}+h)} -f{(\mathfrak{z}_{0})}} {h}$$ exists for every point $$\mathfrak{z}_0$$ in $$\Omega$$, and when h is only allowed to be an invertible hyperbolic number. In this paper we prove that $$\mathbb{D}$$-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 $$\mathbb{D}$$. 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.