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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenWed, 31 Jan 2024 19:53:19 +0000Analytic Functions in General Analysis
https://resolver.caltech.edu/CaltechTHESIS:12132017-090831567
Authors: {'items': [{'id': 'Taylor-Angus-Ellis', 'name': {'family': 'Taylor', 'given': 'Angus Ellis'}, 'show_email': 'NO'}]}
Year: 1936
DOI: 10.7907/V5AY-TP78
<p>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 = z<sub>0</sub> 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.</p>
<p>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.</p>
<p>The study of differentials leads to an important result showing
the relation of the Fréchet and Gateaux concepts of a differential.</p>
<p>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.</p>
<p>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.</p>
<p>To professor A.D. Michal I am indebted for encouragement and advice
at all times.</p>https://thesis.library.caltech.edu/id/eprint/10605