Vol. 18, No. 1, 1966

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Further results in the theory of monodiffric functions

G. J. Kurowski

Vol. 18 (1966), No. 1, 139–147

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
Received: 9 December 1964
Published: 1 July 1966
G. J. Kurowski