Vol. 18, No. 1, 1966

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ISSN: 0030-8730
Further results in the theory of monodiffric functions

G. J. Kurowski

Vol. 18 (1966), No. 1, 139–147
Abstract

This paper considers two fundamental problems in the theory of monodiffric functions; i.e., discrete functions which satisfy the partial difference equation

f(z + 1)− f(z) = − i[f(z + i)− f(z)]

on some region of the discrete z-plane, z = m + in, m = 0,±1,±2, , n = 0,±1,±2, , and which, accordingly, are analogs of analytic functions.

The first problem considered centers about a process analogous to multiplication. A method of analytic extension is presented whereby a function defined along the real axis may be uniquely extended into the upper-half plane as a monodiffric function. The generalized product of two monodiffric functions may then be defined as the extension of a suitable product on the real axis. This definition is shown to be consistent with prior results.

The second problem is concerned with an analog to the Cauchy integral based upon a discrete singularity function which tends to zero as |z| becomes large. The desired singularity function is obtained and the analogous integral formula presented.

Mathematical Subject Classification
Primary: 30.83
Milestones
Received: 9 December 1964
Published: 1 July 1966
Authors
G. J. Kurowski