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

Abstract

In the theory of complex-valued functions of a complex variable, arguably the first striking theorem is that pointwise differentiability implies C ∞ regularity. As mentioned in Ahlfors [Complex Analysis. An Introduction to the Theory of Analytic Functions of One Complex Variable, 3rd edn, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978], there have been a number of studies [Porcelli and Connell, A proof of the power series expansion without Cauchy's formula, Bull. Amer. Math. Soc. 67 (1961), 177-181; Plunkett, A topological proof of the continuity of the derivative of a function of a complex variable, Bull. Amer. Math. Soc. 65 (1959), 1-4] 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.