Pre-functions and Extended pre-functions of Complex Variables

Pre-functions are functions that possess a sequence $\{f_{n}(z,\beta)\}$ which tends to one of the elementary functions as $n$ tends to infinity and $\beta$ tends to 0. The main objective of this paper is to broaden the scope of pre-functions from functions of a real variable to functions of a complex variable by introducing pre-functions of a complex variable. We have analyzed the pre-functions of a complex variable for their properties. The pre-Laguerre, pre-Bessel and pre-Legendre polynomials of a complex variable have been obtained as special cases. Graphs have been used to visualize complex pre-functions.