Differential inclusions, non-absolutely convergent integrals and the first theorem of complex analysis

TL;DR

This paper offers a concise proof of the classical theorem that pointwise differentiability implies smoothness for complex functions, using non-absolutely convergent integrals, and extends it with a broad generalization involving elliptic regularity.

Contribution

It introduces a novel proof technique avoiding complex integration and generalizes the theorem through elliptic regularity concepts.

Findings

Short proof of the theorem using non-absolutely convergent integrals

Generalization of the theorem with elliptic regularity methods

Potential simplification of complex analysis proofs

Abstract

In the theory of complex valued functions of a complex variable arguably the first striking theorem is that pointwise differentiability implies $C^{\infty}$ regularity. As mentioned in Ahlfors's standard textbook there have been a number of studies proving this theorem without use of complex integration but at the cost of considerably more complexity. In this note we will use the theory of non-absolutely convergent integrals to firstly give a very short proof of this result without complex integration and secondly (in combination with some elements of the theory of elliptic regularity) provide a far reaching generalization.