Vol. 3, No. 3, 2010

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A complex finite calculus

Joseph Seaborn and Philip Mummert

Vol. 3 (2010), No. 3, 273–287
Abstract

We explore a complex extension of finite calculus on the integer lattice of the complex plane. f : [i] satisfies the discretized Cauchy–Riemann equations at z if Re(f(z + 1) f(z)) = Im(f(z + i) f(z)) and Re(f(z + i) f(z)) = Im(f(z + 1) f(z)). From this principle arise notions of the discrete path integral, Cauchy’s theorem, the exponential function, discrete analyticity, and falling power series.

Keywords
complex analysis, discrete analytic, finite calculus, finite differences, monodiffric, preholomorphic, Gaussian integers, integer lattice, discrete
Mathematical Subject Classification 2000
Primary: 30G25, 39A12
Milestones
Received: 24 September 2009
Revised: 22 September 2010
Accepted: 23 September 2010
Published: 16 October 2010

Proposed: Johnny Henderson
Communicated by Johnny Henderson
Authors
Joseph Seaborn
Mathematics Department
The University of North Carolina
CB #3250, Phillips Hall 412
Chapel Hill, NC 27599
United States
Philip Mummert
Mathematics Department
Taylor University
236 W Reade Ave
Upland, IN 46989
United States
http://faculty.taylor.edu/phmummert/