Generalized Analytic Functions

672 Pages·1962·27.099 MB·English

by  Vekua I N

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H. H. BEKJ'A
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MOCKBA 1959
GENERALIZED
ANALYTIC FUNCTIONS
by
I. N. VEKUA
ENGLISH TRANSLATION EDITOR
IAN N. SNEDDON
Simson Professor of Mathematics
in the University of Glasgow
PERGAMON PRESS
OXFORD · LONDON · NEW YORK · PARIS
1962
PERGAMON PRESS LTD.
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Copyright © 1962
Pergamon Press Ltd.
Library of Congress Card No. 62-9699
This translation has been made from:—
I. N. Vekua
Obobshchennyye analiticheskiye funktsii
published by Fizmatgiz, Moscow, 1959
Printed in Poland to the order of PWN-Polish Scientific Publishers
by Drukarnia Uniwersytetu Jagiellonskiego, Cracow
ANNOTATION
THIS book is concerned with foundations of the general
theory of generalized analytic functions and some appli-
cations to problems of differential geometry and theory of
shells.
The book is intended for students of advanced courses
of the mechanico-mathematical faculties, postgraduates,
and likewise for research workers.
FOREWORD
TRADITIONAL applications of the classical theory of
analytic functions are mainly connected with the topics
of analysis or its applications based either on the Oauchy-
Biemann system of equations or on equations the solu-
tions of which can comparatively simply be represented
by solutions of the Cauchy-Eiemann system. An example
is provided by the equations of plane hydrodynamics or
of the plane theory of elasticity. Becently, however, the
sphere of applications of the theory of analytic functions
has been considerably extended. In particular, it also
enters into the general theory of elliptic equations. Natu-
rally investigations in this direction were originally con-
cerned with equations with analytic coefficients. In recent
years, however, they have been generalized to equations
with non-analytic coefficients and the results thus ob-
tained make possible a significant development of the
classical theory of analytic functions and its applications.
These generalizations concern a class of functions which
contain families of solutions of a very wide class of elliptic
systems of differential equations of the first order with
two independent variables, and even some functions not
differentiable in the ordinary sense. In this class which
even contains functions non-differentiable in the ordinary
sense, a number of fundamental topological properties of
analytic functions of one complex variable are preserved
(the uniqueness theorem, principle of the argument, etc.).
Moreover, such analytic facts as the Taylor and Laurent
expansions, the Cauchy integral formula, etc. remain valid.
In view of these circumstances the functions under con-
sideration in this book are called generalized analytic
functions.
The first part of the book is concerned with various
problems of the general theory of generalized analytic
xxviii GENERALIZED ANALYTIC FUNCTIONS
functions. The exposition includes not only the foun-
dations of the theory but also a fairly wide range of
boundary value problems. Our considerations are based
on a number of relationships and formulae which connect
the families of solutions of the systems of differential
equations under consideration with the class of analytic
functions of one complex variable. These basic relationships
and formulae constitute the foundations of the entire
theory; they make it possible to reduce investigations to
the classical theory of analytic functions. It should be
observed that the above results constitute a further
natural development of the previous investigations on
equations with analytic coefficients. Just as in the analytic
case the integral representations of solutions contain ker-
nels which depend on the coefficients of the equation.
The constructions carried out make use of integral equa-
tions (over the complex domain) the properties of which
are similar to those of equations of Volterra type employed
in the analytic case.
The power and value of any mathematical theory is
most clearly revealed in comparing its results with the
actual object of investigation. This connection makes it
possible to supply the theory with a definite content,
and moreover, to determine the course of its development.
If the results of a theory enable us to extend considerably
the range of its applications this fact is a sign of the
vitality of the theory. In this respect the possibilities of
the theory of generalized analytic functions are very
large. It is intimately connected with many branches of
analysis, geometry and mechanics (quasi-conformal mapp-
ings, theory of surfaces, theory of shells, gas dynamics, etc.).
For instance, the new analytic structure makes possible
a considerable extension and profound investigation of
geometric and mechanical problems arising in connection
with infinitesimal bendings of surfaces of positive curva-
ture and equilibrium membrane states of stress of convex
shells. These problems are to a large extent considered
FOREWORD XXIX
in the second part of the book; the considerations led to
a number of new results and, moreover, revealed the
geometric and mechanical nature of the generalized
analytic functions.
Unfortunately, it was not possible within the bounds
of this book to present a sufficient exposition of many
other important applications of the theory of generalized
analytic functions. For instance applications to the
problems of quasi-conformal mappings have been dealt
with only very roughly; in this connection important
results were recently obtained by Bojarski [11]. Also,
some applications to non-linear problems are indicated.
Notwithstanding the fact that our reasoning is mainly
based on linear differential equations, the results obtained
can be employed in an investigation of properties of
non-linear elliptic equations.
It should be observed that the book contains many
results of the author and his collaborators published here
for the first time. In addition, it should be noted that
the appendix to Chapter IV was written by B. V. Bojarski.
In the preparation of the manuscript great help was
given to the author by V. S. Vinogradov, L. S. Klabukova,
Sun Che-shen and Ten En Cher. All the figures were
prepared by T. P. Krivenkov. A. V. Bitsadse, B. V. Bo jar-
ski, I. I. Daniluk and E. G. Posnyak read the completed
manuscript of the book and the author is obliged to them
for many valuable suggestions. The author is sincerely
grateful to all those mentioned above.
I. VEKUA
PART ONE
FOUNDATIONS OF THE GENERAL THEORY
OF GENERALIZED ANALYTIC FUNCTIONS
AND BOUNDARY VALUE PROBLEMS
IN THIS part of the book the main attention will be devoted
to the construction of the general theory of complex
functions w(z) of the point z = x + iy, which satisfy an
equation of the form
Bjo + Aw + BW-F {h^\{l+i^). (1.1)
This equation constitutes the complex form of the system
of real equations
dx dy } ' dy ^ da? y v '
The latter system is the canonical form of a more general
elliptic system of equations (Oh. II, §7). A very wide
class of partial differential equations of the second order
can be reduced to a system of the form (1.2) (Ch. Ill, §9).
In the subsequent investigations we shall assume that
the coefficients A and B and the free term F of the equa-
tion (1.1) are summable functions in a power p > 2, in
the domain under consideration. This extension of the
class of the investigated equations is expedient not only
for purely theoretical reasons; it will frequently be ob-
served below that it is also justified from the point of
view of practical applicability.
A theory of such equations cannot, however, be
established by the usual classical methods. To this end