The theory of functions of a complex variable is distinguished from the theory of functions of a real variable by its simplicity - asimplicity is directly traceable to the complexity of the variable.Two of the remarkable simplicities of the theory are, first, that fromthe assumption that f(z) is differentiable throughout the neighborhoodof 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, whichfinds its natural extension in the ideas of Gateaux concerningdifferentials.Much of the work which we present was sketched in a memoir ofGateaux on functionals of continuous functions.* In addition we have developedthe "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-Riemannequations. Nothing has been donewith the implicitly suggested theory of pairs of conjugate harmonic functions, however.
The study of differentials leads to an important result showingthe 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 tosome extent the departures from classical theory which are caused by thegeneralization here undertaken. A more detailed investigation should becarried out in special cases.
I freely express my admiration for the treaties of ProfessorW.F. Osgood, Lehrbuch der Funktionentheorie, to which I have had constantrecourse in the writing of this thesis. Many of the proofs are directlycarried over, with only the slight changes made necessary by the abstractnature of the quantities in hand.
To professor A.D. Michal I am indebted for encouragement and adviceat all times.
