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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.

DIFFERENTIAL INCLUSIONS, NON-ABSOLUTELY CONVERGENT INTEGRALS AND THE FIRST THEOREM OF COMPLEX ANALYSIS