The discrete analogue of a class of entire functions

Abstract

This paper is concerned with complex functions defined at the nodes of a square mesh -Ah that lies in the complex plane and is depicted in Fig. 1.1. The theory of these functions was initiated by R. Isaacs [l-3], J. Ferrand [4], and A. Terracini [5, 61, and has been further developed by R. J. Duffin [7], C. S. Duris [S, 91, E. L. Peterson [lo], A. Washburn [ 111, and J. Rohrer [ 12, 131. The functions investigated are said to be “discrete analytic” and are discrete analogues of analytic functions of a continuous complex variable. A complex function f is discrete analytic on a square S belonging to dh if the difference quotient off across one diagonal of S is equal to the difference quotient off across the other diagonal of S. G. J. Kurowski [14-171 has defined and investigated a class of complex functions that have domain consisting of parallel lines in the complex plane. Functions in this class are said to be “semi-discrete analytic” and are semidiscrete analogues of analytic functions of a continuous complex variable. Semi-discrete analytic functions have many properties in common with discrete analytic functions, but they will not be considered in this paper.