#### 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. $f:ℤ\left[i\right]\to ℂ$ satisfies the discretized Cauchy–Riemann equations at $z$ if $Re\left(f\left(z+1\right)-f\left(z\right)\right)=Im\left(f\left(z+i\right)-f\left(z\right)\right)$ and $Re\left(f\left(z+i\right)-f\left(z\right)\right)=-Im\left(f\left(z+1\right)-f\left(z\right)\right)$. 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