Discrete Analytic Functions Research Articles

Structure of rings of functions with Riemann Stieltjes convolution products

In this note three sets of complex valued functions with pointwise addition and a Riemann Stieltjes convolution product are considered. The functions considered are discrete analytic functions, sequences, and continuous functions of bounded variation defined on the nonnegative real numbers. Each forms a commutative algebra with identity. The discrete analytic functions form a principal ideal ring with five maximal ideals, nine prime ideals, and is essentially a direct sum of four discrete valuation rings. The ring of sequences is isomorphic to an ideal of the ring of discrete analytic functions; it has two maximal and three prime ideals. Both contain divisors of zero. The units, associates, irreducible elements and primes in these two rings are described. The results are used to study the continuous functions; partial results are obtained concerning units and divisors of zero. The product satisfies a convolution theorem.

Further theory of operational calculus on discrete analytic functions

Further theory of operational calculus on discrete analytic functions

A generalization of discrete analytic and harmonic functions

A generalization of discrete analytic and harmonic functions

A convolution product for the solutions of partial difference equations

This paper is concerned with linear partial difference operators L having constant coefficients. The functions considered are defined only on the lattice points of the complex plane. It is shown that with any two solutions of Lu(z) = 0 there is associated a new solution which is represented as a convolution product. This product may be considered as a type of line integral and is based upon a discrete analogue of Green's formula. This development may be regarded as an anology to the pioneering work of H. Lewy concerning the composition of solutions of partial differential equations. It may also be considered a continuation of the investigation pursued by Duffin and Duris in the introduction of a convolution product for discrete analytic functions. HUNT LIBRARY CARNEGIE-MELLON UNIVERSITY A Convolution Product for the Solutions of Partial Difference Equations* by R. J. Duffin and Joan Rohrer CARNEGIE INSTITUTE OF TECHNOLOGY

The discrete analogue of a class of entire functions

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.

Operators of discrete analytic functions and their application

Operators of discrete analytic functions and their application

Operators of discrete analytic functions

Operators of discrete analytic functions

Basic properties of discrete analytic functions

Basic properties of discrete analytic functions