A complex finite calculus

Abstract

We explore a complex extension of finite calculus on the integer lattice of the complex plane. $f:\mathbb{Z}\left[i\right]\to ℂ$ 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.