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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Complex analysis
This book provides a concise survey of the theory of zero
product-determined algebras, which has been developed over the last
15 years. It is divided into three parts. The first part presents
the purely algebraic branch of the theory, the second part presents
the functional analytic branch, and the third part discusses
various applications. The book is intended for researchers and
graduate students in ring theory, Banach algebra theory, and
nonassociative algebra.
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.
Complex Analysis with Mathematica offers a new way of learning and
teaching a subject that lies at the heart of many areas of pure and
applied mathematics, physics, engineering and even art. This book
offers teachers and students an opportunity to learn about complex
numbers in a state-of-the-art computational environment. The
innovative approach also offers insights into many areas too often
neglected in a student treatment, including complex chaos and
mathematical art. Thus readers can also use the book for self-study
and for enrichment. The use of Mathematica enables the author to
cover several topics that are often absent from a traditional
treatment. Students are also led, optionally, into cubic or quartic
equations, investigations of symmetric chaos and advanced conformal
mapping. A CD is included which contains a live version of the
book: in particular all the Mathematica code enables the user to
run computer experiments.
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.
Lester Ford's book was the first treatise in English on automorphic
functions. At the time of its publication (1929), it was welcomed
for its elegant treatment of groups of linear transformations and
for the remarkably clear and explicit exposition throughout the
book. Ford's extraordinary talent for writing has been memorialized
in the prestigious award that bears his name. The book, in the
meantime, has become a recognized classic. Ford's approach is
primarily through analytic function theory. The first part of the
book covers groups of linear transformations, especially Fuchsian
groups, fundamental domains, and functions that are invariant under
the groups, including the classical elliptic modular functions and
Poincare theta series. The second part of the book covers conformal
mappings, uniformization, and connections between automorphic
functions and differential equations with regular singular points,
such as the hypergeometric equation.
This fine book by Herb Clemens quickly became a favorite of many
algebraic geometers when it was first published in 1980. It has
been popular with novices and experts ever since. It is written as
a book of 'impressions' of a journey through the theory of complex
algebraic curves. Many topics of compelling beauty occur along the
way. A cursory glance at the subjects visited reveals a wonderfully
eclectic selection, from conics and cubics to theta functions,
Jacobians, and questions of moduli. By the end of the book, the
theme of theta functions becomes clear, culminating in the Schottky
problem. The author's intent was to motivate further study and to
stimulate mathematical activity. The attentive reader will learn
much about complex algebraic curves and the tools used to study
them. The book can be especially useful to anyone preparing a
course on the topic of complex curves or anyone interested in
supplementing his/her reading.
Presents Real & Complex Analysis Together Using a Unified
Approach
A two-semester course in analysis at the advanced undergraduate or
first-year graduate level
Unlike other undergraduate-level texts, Real and Complex
Analysis develops both the real and complex theory together. It
takes a unified, elegant approach to the theory that is consistent
with the recommendations of the MAA s 2004 Curriculum Guide.
By presenting real and complex analysis together, the authors
illustrate the connections and differences between these two
branches of analysis right from the beginning. This combined
development also allows for a more streamlined approach to real and
complex function theory. Enhanced by more than 1,000 exercises, the
text covers all the essential topics usually found in separate
treatments of real analysis and complex analysis. Ancillary
materials are available on the book s website.
This book offers a unique, comprehensive presentation of both
real and complex analysis. Consequently, students will no longer
have to use two separate textbooks one for real function theory and
one for complex function theory.
Intractability is a growing concern across the cognitive sciences:
while many models of cognition can describe and predict human
behavior in the lab, it remains unclear how these models can scale
to situations of real-world complexity. Cognition and
Intractability is the first book to provide an accessible
introduction to computational complexity analysis and its
application to questions of intractability in cognitive science.
Covering both classical and parameterized complexity analysis, it
introduces the mathematical concepts and proof techniques that can
be used to test one's intuition of (in)tractability. It also
describes how these tools can be applied to cognitive modeling to
deal with intractability, and its ramifications, in a systematic
way. Aimed at students and researchers in philosophy, cognitive
neuroscience, psychology, artificial intelligence, and linguistics
who want to build a firm understanding of intractability and its
implications in their modeling work, it is an ideal resource for
teaching or self-study.
This book provides a thorough introduction to the theory of complex
semisimple quantum groups, that is, Drinfeld doubles of
q-deformations of compact semisimple Lie groups. The presentation
is comprehensive, beginning with background information on Hopf
algebras, and ending with the classification of admissible
representations of the q-deformation of a complex semisimple Lie
group. The main components are: - a thorough introduction to
quantized universal enveloping algebras over general base fields
and generic deformation parameters, including finite dimensional
representation theory, the Poincare-Birkhoff-Witt Theorem, the
locally finite part, and the Harish-Chandra homomorphism, - the
analytic theory of quantized complex semisimple Lie groups in terms
of quantized algebras of functions and their duals, - algebraic
representation theory in terms of category O, and - analytic
representation theory of quantized complex semisimple groups. Given
its scope, the book will be a valuable resource for both graduate
students and researchers in the area of quantum groups.
Analysis underpins calculus, much as calculus underpins virtually
all mathematical sciences. A sound understanding of analysis'
results and techniques is therefore valuable for a wide range of
disciplines both within mathematics itself and beyond its
traditional boundaries. This text seeks to develop such an
understanding for undergraduate students on mathematics and
mathematically related programmes. Keenly aware of contemporary
students' diversity of motivation, background knowledge and time
pressures, it consistently strives to blend beneficial aspects of
the workbook, the formal teaching text, and the informal and
intuitive tutorial discussion. The authors devote ample space and
time for development of confidence in handling the fundamental
ideas of the topic. They also focus on learning through doing,
presenting a comprehensive range of examples and exercises, some
worked through in full detail, some supported by sketch solutions
and hints, some left open to the reader's initiative. Without
undervaluing the absolute necessity of secure logical argument,
they legitimise the use of informal, heuristic, even imprecise
initial explorations of problems aimed at deciding how to tackle
them. In this respect they authors create an atmosphere like that
of an apprenticeship, in which the trainee analyst can look over
the shoulder of the experienced practitioner.
A world model: economies, trade, migration, security and
development aid. This bookprovides the analytical capability to
understand and explore the dynamics of globalisation. It is
anchored in economic input-output models of over 200 countries and
their relationships through trade, migration, security and
development aid. The tools of complexity science are brought to
bear and mathematical and computer models are developed both for
the elements and for an integrated whole. Models are developed at a
variety of scales ranging from the global and international trade
through a European model of inter-sub-regional migration to piracy
in the Gulf and the London riots of 2011. The models embrace the
changing technology of international shipping, the impacts of
migration on economic development along with changing patterns of
military expenditure and development aid. A unique contribution is
the level of spatial disaggregation which presents each of 200+
countries and their mutual interdependencies along with some finer
scale analyses of cities and regions. This is the first global
model which offers this depth of detail with fully work-out models,
these provide tools for policy making at national, European and
global scales. Global dynamics: * Presents in depth models of
global dynamics. * Provides a world economic model of 200+
countries and their interactions through trade, migration, security
and development aid. * Provides pointers to the deployment of
analytical capability through modelling in policy development. *
Features a variety of models that constitute a formidable toolkit
for analysis and policy development. * Offers a demonstration of
the practicalities of complexity science concepts. This book is for
practitioners and policy analysts as well as those interested in
mathematical model building and complexity science as well as
advanced undergraduate and postgraduate level students.
The central theme of this reference book is the metric geometry of
complex analysis in several variables. Bridging a gap in the
current literature, the text focuses on the fine behavior of the
Kobayashi metric of complex manifolds and its relationships to
dynamical systems, hyperbolicity in the sense of Gromov and
operator theory, all very active areas of research. The modern
points of view expressed in these notes, collected here for the
first time, will be of interest to academics working in the fields
of several complex variables and metric geometry. The different
topics are treated coherently and include expository presentations
of the relevant tools, techniques and objects, which will be
particularly useful for graduate and PhD students specializing in
the area.
Chaos and Dynamical Systems presents an accessible, clear
introduction to dynamical systems and chaos theory, important and
exciting areas that have shaped many scientific fields. While the
rules governing dynamical systems are well-specified and simple,
the behavior of many dynamical systems is remarkably complex. Of
particular note, simple deterministic dynamical systems produce
output that appears random and for which long-term prediction is
impossible. Using little math beyond basic algebra, David Feldman
gives readers a grounded, concrete, and concise overview. In
initial chapters, Feldman introduces iterated functions and
differential equations. He then surveys the key concepts and
results to emerge from dynamical systems: chaos and the butterfly
effect, deterministic randomness, bifurcations, universality, phase
space, and strange attractors. Throughout, Feldman examines
possible scientific implications of these phenomena for the study
of complex systems, highlighting the relationships between
simplicity and complexity, order and disorder. Filling the gap
between popular accounts of dynamical systems and chaos and
textbooks aimed at physicists and mathematicians, Chaos and
Dynamical Systems will be highly useful not only to students at the
undergraduate and advanced levels, but also to researchers in the
natural, social, and biological sciences.
This second edition presents a collection of exercises on the
theory of analytic functions, including completed and detailed
solutions. It introduces students to various applications and
aspects of the theory of analytic functions not always touched on
in a first course, while also addressing topics of interest to
electrical engineering students (e.g., the realization of rational
functions and its connections to the theory of linear systems and
state space representations of such systems). It provides examples
of important Hilbert spaces of analytic functions (in particular
the Hardy space and the Fock space), and also includes a section
reviewing essential aspects of topology, functional analysis and
Lebesgue integration. Benefits of the 2nd edition Rational
functions are now covered in a separate chapter. Further, the
section on conformal mappings has been expanded.
This book discusses the complex theory of differential equations or
more precisely, the theory of differential equations on
complex-analytic manifolds. Although the theory of differential
equations on real manifolds is well known - it is described in
thousands of papers and its usefulness requires no comments or
explanations - to date specialists on differential equations have
not focused on the complex theory of partial differential
equations. However, as well as being remarkably beautiful, this
theory can be used to solve a number of problems in real theory,
for instance, the Poincare balayage problem and the mother body
problem in geophysics. The monograph does not require readers to be
familiar with advanced notions in complex analysis, differential
equations, or topology. With its numerous examples and exercises,
it appeals to advanced undergraduate and graduate students, and
also to researchers wanting to familiarize themselves with the
subject.
This volume collects lecture notes from courses offered at
several conferences and workshops, and provides the first
exposition in book form of the basic theory of the Kahler-Ricci
flow and its current state-of-the-art. While several excellent
books on Kahler-Einstein geometry are available, there have been no
such works on the Kahler-Ricci flow. The book will serve as a
valuable resource for graduate students and researchers in complex
differential geometry, complex algebraic geometry and Riemannian
geometry, and will hopefully foster further developments in this
fascinating area of research.
The Ricci flow was first introduced by R. Hamilton in the early
1980s, and is central in G. Perelman's celebrated proof of the
Poincare conjecture. When specialized for Kahler manifolds, it
becomes the Kahler-Ricci flow, and reduces to a scalar PDE
(parabolic complex Monge-Ampere equation).
As a spin-off of his breakthrough, G. Perelman proved the
convergence of the Kahler-Ricci flow on Kahler-Einstein manifolds
of positive scalar curvature (Fano manifolds). Shortly after, G.
Tian and J. Song discovered a complex analogue of Perelman's ideas:
the Kahler-Ricci flow is a metric embodiment of the Minimal Model
Program of the underlying manifold, and flips and divisorial
contractions assume the role of Perelman's surgeries."
Now in its fourth edition, the first part of this book is devoted to the basic material of complex analysis, while the second covers many special topics, such as the Riemann Mapping Theorem, the gamma function, and analytic continuation. Power series methods are used more systematically than is found in other texts, and the resulting proofs often shed more light on the results than the standard proofs. While the first part is suitable for an introductory course at undergraduate level, the additional topics covered in the second part give the instructor of a gradute course a great deal of flexibility in structuring a more advanced course.
The aim of this book is to provide a comprehensive account of
higher dimensional Nevanlinna theory and its relations with
Diophantine approximation theory for graduate students and
interested researchers. This book with nine chapters systematically
describes Nevanlinna theory of meromorphic maps between algebraic
varieties or complex spaces, building up from the classical theory
of meromorphic functions on the complex plane with full proofs in
Chap. 1 to the current state of research. Chapter 2 presents the
First Main Theorem for coherent ideal sheaves in a very general
form. With the preparation of plurisubharmonic functions, how the
theory to be generalized in a higher dimension is described. In
Chap. 3 the Second Main Theorem for differentiably non-degenerate
meromorphic maps by Griffiths and others is proved as a prototype
of higher dimensional Nevanlinna theory. Establishing such a Second
Main Theorem for entire curves in general complex algebraic
varieties is a wide-open problem. In Chap. 4, the Cartan-Nochka
Second Main Theorem in the linear projective case and the
Logarithmic Bloch-Ochiai Theorem in the case of general algebraic
varieties are proved. Then the theory of entire curves in
semi-abelian varieties, including the Second Main Theorem of
Noguchi-Winkelmann-Yamanoi, is dealt with in full details in Chap.
6. For that purpose Chap. 5 is devoted to the notion of
semi-abelian varieties. The result leads to a number of
applications. With these results, the Kobayashi hyperbolicity
problems are discussed in Chap. 7. In the last two chapters
Diophantine approximation theory is dealt with from the viewpoint
of higher dimensional Nevanlinna theory, and the Lang-Vojta
conjecture is confirmed in some cases. In Chap. 8 the theory over
function fields is discussed. Finally, in Chap. 9, the theorems of
Roth, Schmidt, Faltings, and Vojta over number fields are presented
and formulated in view of Nevanlinna theory with results motivated
by those in Chaps. 4, 6, and 7.
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