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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)]](a130x.png)
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.
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