|
Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Complex analysis
Since the appearance of Kobayashi's book, there have been several
re sults at the basic level of hyperbolic spaces, for instance
Brody's theorem, and results of Green, Kiernan, Kobayashi, Noguchi,
etc. which make it worthwhile to have a systematic exposition.
Although of necessity I re produce some theorems from Kobayashi, I
take a different direction, with different applications in mind, so
the present book does not super sede Kobayashi's. My interest in
these matters stems from their relations with diophan tine
geometry. Indeed, if X is a projective variety over the complex
numbers, then I conjecture that X is hyperbolic if and only if X
has only a finite number of rational points in every finitely
generated field over the rational numbers. There are also a number
of subsidiary conjectures related to this one. These conjectures
are qualitative. Vojta has made quantitative conjectures by
relating the Second Main Theorem of Nevan linna theory to the
theory of heights, and he has conjectured bounds on heights
stemming from inequalities having to do with diophantine
approximations and implying both classical and modern conjectures.
Noguchi has looked at the function field case and made substantial
progress, after the line started by Grauert and Grauert-Reckziegel
and continued by a recent paper of Riebesehl. The book is divided
into three main parts: the basic complex analytic theory,
differential geometric aspects, and Nevanlinna theory. Several
chapters of this book are logically independent of each other."
The theory of General Relativity, after its invention by Albert
Einstein, remained for many years a monument of mathemati cal
speculation, striking in its ambition and its formal beauty, but
quite separated from the main stream of modern Physics, which had
centered, after the early twenties, on quantum mechanics and its
applications. In the last ten or fifteen years, however, the
situation has changed radically. First, a great deal of significant
exper en tal data became available. Then important contributions
were made to the incorporation of general relativity into the
framework of quantum theory. Finally, in the last three years,
exciting devel opments took place which have placed general
relativity, and all the concepts behind it, at the center of our
understanding of par ticle physics and quantum field theory.
Firstly, this is due to the fact that general relativity is really
the "original non-abe lian gauge theory," and that our description
of quantum field in teractions makes extensive use of the concept
of gauge invariance. Secondly, the ideas of supersymmetry have
enabled theoreticians to combine gravity with other elementary
particle interactions, and to construct what is perhaps the first
approach to a more finite quantum theory of gravitation, which is
known as super gravity."
In 1960 Wilhelm Stoll joined the University of Notre Dame faculty
as Professor of Mathematics, and in October, 1984 the university
acknowledged his many years of distinguished service by holding a
conference in complex analysis in his honour. This volume is the
proceedings of that conference. It was our priviledge to serve,
along with Nancy K. Stanton, as conference organizers. We are
grateful to the College of Science of the University of Notre Dame
and to the National Science Foundation for their support. In the
course of a career that has included the publication of over sixty
research articles and the supervision of eighteen doctoral
students, Wilhelm Stoll has won the affection and respect of his
colleagues for his diligence, integrity and humaneness. The
influence of his ideas and insights and the subsequent
investigations they have inspired is attested to by several of the
articles in the volume. On behalf of the conference partipants and
contributors to this volume, we wish Wilhelm Stoll many more years
of happy and devoted service to mathematics. Alan Howard Pit-Mann
Wong VII III c: ... c: o U CI> .r. .... o e:: J o a:: a.:: J o
... (. : J VIII '" Q) g> a. '" Q) E z '" ..... o Q) E Q) ..c eX
IX Participants on the Group Picture Qi-keng LU, Professor, Chinese
Academy of Science, Peking, China.
This book gives an introductory exposition of the theory of
hyperfunctions and regular singularities. This first English
introduction to hyperfunctions brings readers to the forefront of
research in the theory of harmonic analysis on symmetric spaces. A
substantial bibliography is also included. This volume is based on
a paper which was awarded the 1983 University of Copenhagen Gold
Medal Prize.
In recent years there has been increasing interaction among various
branches of mathematics. This is especially evident in the theory
of several complex variables where fruitful interplays of the
methods of algebraic geometry, differential geometry, and partial
differential equations have led to unexpected insights and new
directions of research. In China there has been a long tradition of
study in complex analysis, differential geometry and differential
equations as interrelated subjects due to the influence of
Professors S. S. Chern and L. K. Hua. After a long period of
isolation, in recent years there is a resurgence of scientific
activity and a resumption of scientific exchange with other
countries. The Hangzhou conference is the first international
conference in several complex variables held in China. It offered a
good opportunity for mathematicians from China, U.S., Germany,
Japan, Canada, and France to meet and to discuss their work. The
papers presented in the conference encompass all major aspects of
several complex variables, in particular, in such areas as complex
differential geometry, integral representation, boundary behavior
of holomorphic functions, invariant metrics, holomorphic vector
bundles, and pseudoconvexity. Most of the participants wrote up
their talks for these proceedings. Some of the papers are surveys
and the others present original results. This volume constitutes an
overview of the current trends of research in several complex
variables.
This contributed volume collects papers based on courses and talks
given at the 2017 CIMPA school Harmonic Analysis, Geometric Measure
Theory and Applications, which took place at the University of
Buenos Aires in August 2017. These articles highlight recent
breakthroughs in both harmonic analysis and geometric measure
theory, particularly focusing on their impact on image and signal
processing. The wide range of expertise present in these articles
will help readers contextualize how these breakthroughs have been
instrumental in resolving deep theoretical problems. Some topics
covered include: Gabor frames Falconer distance problem Hausdorff
dimension Sparse inequalities Fractional Brownian motion Fourier
analysis in geometric measure theory This volume is ideal for
applied and pure mathematicians interested in the areas of image
and signal processing. Electrical engineers and statisticians
studying these fields will also find this to be a valuable
resource.
This book is first of all designed as a text for the course usually
called "theory of functions of a real variable". This course is at
present cus tomarily offered as a first or second year graduate
course in United States universities, although there are signs that
this sort of analysis will soon penetrate upper division
undergraduate curricula. We have included every topic that we think
essential for the training of analysts, and we have also gone down
a number of interesting bypaths. We hope too that the book will be
useful as a reference for mature mathematicians and other
scientific workers. Hence we have presented very general and
complete versions of a number of important theorems and
constructions. Since these sophisticated versions may be difficult
for the beginner, we have given elementary avatars of all important
theorems, with appro priate suggestions for skipping. We have given
complete definitions, ex planations, and proofs throughout, so that
the book should be usable for individual study as well as for a
course text. Prerequisites for reading the book are the following.
The reader is assumed to know elementary analysis as the subject is
set forth, for example, in TOM M. ApOSTOL'S Mathematical Analysis
[Addison-Wesley Publ. Co., Reading, Mass., 1957], or WALTER RUDIN'S
Principles of Mathe nd matical Analysis [2 Ed., McGraw-Hill Book
Co., New York, 1964].
to Classical Complex Analysis Vol. 1 by Robert B. Burckel Kansas
State University 1979 BIRKHAUSER VERLAG BASEL UND STUTTGART
CIP-Kurztitelaufnahme der Deutschen Bibliothek Burckel, Robert B.:
An introduction to classical complex analysis I by Robert B.
Burckel. - Basel. Stuttgart: Birkhiiuser. Vol. I. - 1979.
(Lehrbilcher und Monographien aus dem Gebiete der exakten
Wissenschaften: Math. Reihe; Bd. 64) All Rights Reserved. No part
of this publication may be reproduced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic,
mechanical, photocopying, recording or otherwise, without the prior
permission of the Copyright owner. (c) Birkhiiuser Verlag Basel,
1979 North and South America Edition published by ACADEMIC PRESS.
INC. III Fifth Avenue, New York, New York 10003 (Pure and Applied
Mathematics, A Series of Monographs and Textbooks, Volume 82)
ISBN-13: 978-3-0348-9376-3 e-ISBN-13: 978-3-0348-9374-9 DOl:
10.1007/978-3-0348-9374-9 Library of Congress Catalog Card Number
78-67403 5 Contents Volume I PREFACE 9 Chapter 0 PREREQUISITES AND
PRELIMINARIES 13 1 Set Theory 13 2 Algebra 14 3 The Battlefield 14
4 Metric Spaces 15 5 Limsup and All That 18 6 Continuous Functions
20 7 Calculus 21 Chapter I CURVES, CONNECTEDNESS AND CONVEXITY 22 1
Elementary Results on Connectedness 22 2 Connectedness of
Intervals, Curves and Convex Sets 23 3 The Basic Connectedness
Lemma 28 4 Components and Compact Exhaustions 29 5 Connectivity of
a Set 33 6 Extension Theorems 37 Notes to Chapter I 39"
This book is intended for someone learning functions of a complex
variable and who enjoys using MATLAB. It will enhance the exprience
of learning complex variable theory and will strengthen the
knowledge of someone already trained in ths branch of advanced
calculus. ABET, the accrediting board for engineering programs,
makes it clear that engineering graduates must be skilled in the
art of programming in a language such as MATLAB (R). Supplying
students with a bridge between the functions of complex variable
theory and MATLAB, this supplemental text enables instructors to
easily add a MATLAB component to their complex variables courses. A
MATLAB (R) Companion to Complex Variables provides readers with a
clear understanding of the utility of MATLAB in complex variable
calculus. An ideal adjunct to standard texts on the functions of
complex variables, the book allows professors to quickly find and
assign MATLAB programming problems that will strengthen students'
knowledge of the language and concepts of complex variable theory.
The book shows students how MATLAB can be a powerful learning aid
in such staples of complex variable theory as conformal mapping,
infinite series, contour integration, and Laplace and Fourier
transforms. In addition to MATLAB programming problems, the text
includes many examples in each chapter along with MATLAB code.
Fractals, the most recent interesting topic involving complex
variables, demands to be treated with a language such as MATLAB.
This book concludes with a Coda, which is devoted entirely to this
visually intriguing subject. MATLAB is not without constraints,
limitations, irritations, and quirks, and there are subtleties
involved in performing the calculus of complex variable theory with
this language. Without knowledge of these subtleties, engineers or
scientists attempting to use MATLAB for solutions of practical
problems in complex variable theory suffer the risk of making major
mistakes. This book serves as an early warning system about these
pitfalls.
This book contains a rigorous coverage of those topics (and only
those topics) that, in the author's judgement, are suitable for
inclusion in a first course on Complex Functions. Roughly speaking,
these can be summarized as being the things that can be done with
Cauchy's integral formula and the residue theorem. On the
theoretical side, this includes the basic core of the theory of
differentiable complex functions, a theory which is unsurpassed in
Mathematics for its cohesion, elegance and wealth of surprises. On
the practical side, it includes the computational applications of
the residue theorem. Some prominence is given to the latter,
because for the more sceptical student they provide the
justification for inventing the complex numbers. Analytic
continuation and Riemann surfaces form an essentially different
chapter of Complex Analysis. A proper treatment is far too
sophisticated for a first course, and they are therefore excluded.
The aim has been to produce the simplest possible rigorous
treatment of the topics discussed. For the programme outlined
above, it is quite sufficient to prove Cauchy'S integral theorem
for paths in star-shaped open sets, so this is done. No form of the
Jordan curve theorem is used anywhere in the book.
This book is a continuation of Volume I of the same title [Grund
lehren der mathematischen Wissenschaften, Band 115 ]. We constantly
1 1. The textbook Real and cite definitions and results from Volume
abstract analysis by E. HEWITT and K. R. STROMBERG [Berlin * Gottin
gen *Heidelberg: Springer-Verlag 1965], which appeared between the
publication of the two volumes of this work, contains many standard
facts from analysis. We use this book as a convenient reference for
such facts, and denote it in the text by RAAA. Most readers will
have only occasional need actually to read in RAAA. Our goal in
this volume is to present the most important parts of harmonic
analysis on compact groups and on locally compact Abelian groups.
We deal with general locally compact groups only where they are the
natural setting for what we are considering, or where one or
another group provides a useful counterexample. Readers who are
interested only in compact groups may read as follows: 27, Appendix
D, 28-30 [omitting subheads (30.6)-(30.60)ifdesired],
(31.22)-(31.25), 32, 34-38, 44. Readers who are interested only in
locally compact Abelian groups may read as follows: 31-33, 39-42,
selected Mis cellaneous Theorems and Examples in 34-38. For all
readers, 43 is interesting but optional. Obviously we have not been
able to cover all of harmonic analysis.
This monograph examines rotation sets under the multiplication by d
(mod 1) map and their relation to degree d polynomial maps of the
complex plane. These sets are higher-degree analogs of the
corresponding sets under the angle-doubling map of the circle,
which played a key role in Douady and Hubbard's work on the
quadratic family and the Mandelbrot set. Presenting the first
systematic study of rotation sets, treating both rational and
irrational cases in a unified fashion, the text includes several
new results on their structure, their gap dynamics, maximal and
minimal sets, rigidity, and continuous dependence on parameters.
This abstract material is supplemented by concrete examples which
explain how rotation sets arise in the dynamical plane of complex
polynomial maps and how suitable parameter spaces of such
polynomials provide a complete catalog of all such sets of a given
degree. As a main illustration, the link between rotation sets of
degree 3 and one-dimensional families of cubic polynomials with a
persistent indifferent fixed point is outlined. The monograph will
benefit graduate students as well as researchers in the area of
holomorphic dynamics and related fields.
|
|