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 [Ah 78] there have been a number of studies [Po 61], [Pl 59] 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. One of the first and most striking theorems about the analysis of complex valued functions of a complex variable is that merely from considering the class of pointwise complex differentiable functions on an open set we instantly find ourself in the category of C∞ functions. Theorem 1. Given open set Ω ⊂ C. Suppose f : Ω → C is a complex differentiable at every point. Then f is C∞ on Ω. Typically Theorem 1 is proved via the method of complex integration. The first step is to prove that the integral of a differentiable function over the boundary of a rectangle inside a ball is zero, this was first proved by Goursat [Go 01]. The existence of an anti-derivative is then concluded, Cauchy’s integral formula follows and it is shown that you can differentiate through the integral of Cauchy’s integral formula infinitely many times and hence the function is C∞. On the first paragraph of page 101 of Ahlfors’s standard text [Ah 78], he writes that many important properties of analytic functions are difficult to prove without use of complex integration. Ahlfors states that only recently1 it has been possible to prove continuity of the gradient (or the existence of higher gradients) without the use of complex integration. He refers to articles of Plunkett [Pl 59], and Porcelli and Connell [Po 61] both of which rely on a topological theorem of Whyburn [Wh 58]. Ahlfors notes that both these proofs are much more complicated than the original proof. It turns out that generalizations of Goursat’s theorem have a long history and one line of generalization provides an alternative proof of the theorem that in essence does not require complex integration. This line of research was started by Montel [Mo 07] and further developed by Looman [Lo 23] and Menchoff [Me 36] (their theorem receives a very clear exposition in Saks [Sa 37]) and later by Tolstov [To 42]. Although not explicitly stated in these works the method of proof was essentially to construct a Denjoy type integral to integrate the divergence of a differentiable vector field [Pr 12]. The explicit application of the theory of non-absolutely convergent integrals appears to have first been made in [Ju-No 90] by Jurkat and Nonnenmacher, they used a kind of Perron integral in the plane to prove a generalization of Goursat’s theorem due to Besicovitch [Be 31]. The purpose of this note is firstly to use the theory of non-absolutely convergent integrals to provide the shortest proof of Theorem 1, the proof we provide is also independent of the theory of complex line integrals. Secondly by rephrasing Theorem 1 in terms of differential inclusions 2010 Mathematics Subject Classification. 30A99,26A39,35J47.