The classification of complex generalized Riemann derivatives

Abstract

This article completes the more than a half a century old problem of finding the equivalences between generalized Riemann derivatives. The real functions case is studied in a recent paper by the authors. The complex functions case developed here is more general and comes with numerous applications.We say that a complex generalized Riemann derivative A implies another complex generalized Riemann derivative B if whenever a measurable complex function is A-differentiable at z then it is B-differentiable at z. We characterize all pairs (ΔA,ΔB) of complex generalized Riemann differences of any orders for which A-differentiability implies B-differentiability, and those for which A-differentiability is equivalent to B-differentiability. We show that all m points based generalized Riemann difference quotients of order n that Taylor approximate the ordinary nth derivative to highest rank form a projective variety of dimension m−n for which an explicit parametrization is given.One application provides an infinite number of equivalent ways to define analyticity. For example, a function f is analytic on a region Ω if and only if at each z in Ω, the limitlimh→0⁡f(z+h)+f(z+ih)−f(z−h)+f(z−ih)−2f(z)2h exists and is a finite number. Four more applications relate the classification of complex generalized Riemann derivatives to analyticity and the Cauchy-Riemann equations, and to the theory of best approximations.