The discrete analogue of a class of entire functions

Discrete analytic functions are coinplex valued functions defined at points of the z-plane with integer coordinates. The real and imaginary parts of these functions are required to satisfy difference equations analogous to the Cauchy-Riemann equations. The pseudo power z() is a discrete analytic function which is asymptotic to the ordinary power z for large z. The central topic of this paper is the correspondence between the pseudo power series f =/L cnz( ) and the ordinary power series F =£~ cnz . The coefficients are restricted by the relation lim sup | n! c n | *<C 2 and this insures that both series converge at all points. The correspondence defines a linear transformation T such that f = TF. It is shown that T can be expressed as a contour integral and that T has a unique inverse. By virtue of the transformation various operations on the entire function class (F) induce corresponding operations on the discrete function class (f). In particular a ring of discrete analytic functions can be formed by defining the product of the functions f and g as T(FG). 3 HUNT LIBRARY s CARNEGIE-MELLON UNIVERSE The Discrete Analogue of a Class of Entire Functions R. J. Duffin and Elinor L. Peterson Department of Mathematics, Carnegie Institute of Technology, Pittsburgh 13, Pennsylvania. This author was partially supported by Research Grant DA-AROD-31-124-G78 of the U.S. Army Research Office Durham. Department of Mathematics, University of Michigan, Ann Arbor, Michigan. Research reported here constituted part of this authors Ph.D. thesis, which was written at Carnegie Institute of Technology under the guidance of R. J. Duffin, with financial support from a Socony Mobil Fellowship and Research Grant DA-AROD-31-124-G78 of the U.S. Army Research Office Durham.