Analytic functions of a generalized complex variable

In this dissertation we consider a generalized complex space Cq whose elements are of the form x + qy, where x,y∈R and q=a+bi∈C,b≠0. Following the work of Capelli (1940) we develop an algebra on Cq along with notions of modulus, conjugate, and argument. Functions with domain and range in Cq are then considered along with their derivatives and integrals. An version of Cauchy's Theorem for Cq is established and from this we develop analogs of many of the classical theorems in complex analysis.