Analytic Functions in General Analysis

Citation

Taylor, Angus Ellis (1936) Analytic Functions in General Analysis. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/V5AY-TP78. https://resolver.caltech.edu/CaltechTHESIS:12132017-090831567

Abstract

The theory of functions of a complex variable is distinguished from the theory of functions of a real variable by its simplicity - a simplicity is directly traceable to the complexity of the variable. Two of the remarkable simplicities of the theory are, first, that from the assumption that f(z) is differentiable throughout the neighborhood of a point z = z₀ follows the existence of all higher derivatives and the convergence of the Taylor's series for f(z); and secondly, that we are able to classify in simple terms the possible singularities of an analytic function.

It is the purpose of this work to generalize, insofar as is possible, the basic theorems of the classical theory, and to investigate in what measure the simplicities mentioned above are preserved when the arguments and function values lie in a Banach space. Of the three principally recognized points of view which are used in developing the theory of analytic functions we have used mainly the one due to Cauchy, which finds its natural extension in the ideas of Gateaux concerning differentials. Much of the work which we present was sketched in a memoir of Gateaux on functionals of continuous functions.* In addition we have developed the "Weierstrassian" properties of analytic functions, using as a foundation the notion of polynomial as set forth by R.S. Martin.** Finally, a brief section is devoted to a generalization of the Cauchy-Riemann equations. Nothing has been done with the implicitly suggested theory of pairs of conjugate harmonic functions, however.

The study of differentials leads to an important result showing the relation of the Fréchet and Gateaux concepts of a differential.

The classification of singular points is a most difficult problem. We have dealt completely with removable singularities, and showed to some extent the departures from classical theory which are caused by the generalization here undertaken. A more detailed investigation should be carried out in special cases.

I freely express my admiration for the treaties of Professor W.F. Osgood, Lehrbuch der Funktionentheorie, to which I have had constant recourse in the writing of this thesis. Many of the proofs are directly carried over, with only the slight changes made necessary by the abstract nature of the quantities in hand.

To professor A.D. Michal I am indebted for encouragement and advice at all times.

Thesis Files