Higher Derivatives of Analytic Functions From the Standpoint of Topological Analysis

Abstract

One of the remarkable features of classical complex variable theory is the indispensable part played by integration. As L. V. Ahlfors has emphasised in his Complex Analysis [1], many fundamental properties of analytic functions, such as the existence of higher derivatives, have never been proved without resort to complex integrals or equivalent toolsf. The elementary proof by Eggleston and Ursell [2] that an analytic function is light and open makes certain important theorems, including those of Rouche and Hurwitz, independent of integration [3], and has led more recently to a topological proof of the continuity of the derivative of an analytic function [4]. Other fundamental properties of analytic functions have remained inaccessible by methods of this kind. The present paper gives a development of complex variable theory which leads in particular to the existence of the higher derivatives and to the Taylor exj)ansion without any appeal to complex integration. It will be evident that other standard results can be derived directly from the theorems here proved.Our starting point is a form of the maximum principle used by Whyburn ([1], p. 74) in his presentation of the above-mentioned theorem of Eggleston and Ursell. An adequate form of this result is the following, and in this limited form it can also be proved, with less topology, by keeping closer to the treatment in [2]. LEMMA A. Iff(z) is continuous inside and on a circle C.ivith interior R, if it is differentiate in R except at a finite number of points, and if \f(z)\ ^ M on C, then \f(z)\ < M in R.By applying this result to the function g{z)/z we immediately obtain Schwarz's Lemma:If y( z ) * 5 differentiate in | z | < r with g(0) = 0 and \g(z)\^.M, then | g{z)/z\ < M/r when 0 < | z\ < r. This is sufficient to validate the fundamental theorem on normal families of analytic functions:A family of functions differentiate and locally uniformly bounded in a region R is normal in R.To prove this it will be enough to establish that the family is equicontinuous on every compact subset of R, for once this has been achieved Received 16 July, 1960. f While this paper was in the press non-integrational proofs of some of these properties were announced by P. Porcelli and E.

Complex derivatives are continuous - three self-contained proofs. Part 1

“…Plunkett [34] gave an integration-free proof of continuity of complex derivative in 1959 but was unaware that this had been already achieved by Adel'son-Vel'skii and Kronrod [3] in the USSR 15 years earlier, as pointed out by Lorent [30]; we will discuss their proof in part 2. Topological techniques developed by Whyburn [44] were used by him [45], Connell [13,14], Connell and Porcelli [15,36], Plunkett [34,35] and Read [37] to build the theory of analytic functions in an integration-free way on a topological base. Eggleston and Ursell [18] also deduced properties of analytic functions like the maximum modulus principle by the winding number.…”