Complex Analysis

Author: Elias M. Stein,Rami Shakarchi

Publisher: Princeton University Press

ISBN: 1400831156

Category: Mathematics

Page: 400

View: 6776

With this second volume, we enter the intriguing world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex. From there, one proceeds to the main properties of holomorphic functions, whose proofs are generally short and quite illuminating: the Cauchy theorems, residues, analytic continuation, the argument principle. With this background, the reader is ready to learn a wealth of additional material connecting the subject with other areas of mathematics: the Fourier transform treated by contour integration, the zeta function and the prime number theorem, and an introduction to elliptic functions culminating in their application to combinatorics and number theory. Thoroughly developing a subject with many ramifications, while striking a careful balance between conceptual insights and the technical underpinnings of rigorous analysis, Complex Analysis will be welcomed by students of mathematics, physics, engineering and other sciences. The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Complex Analysis is the second, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.
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Real Analysis

Measure Theory, Integration, and Hilbert Spaces

Author: Elias M. Stein,Rami Shakarchi

Publisher: Princeton University Press

ISBN: 1400835569

Category: Mathematics

Page: 424

View: 931

Real Analysis is the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science. After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets. Each chapter has a series of exercises, from the relatively easy to the more complex, that are tied directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises. As with the other volumes in the series, Real Analysis is accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels. Also available, the first two volumes in the Princeton Lectures in Analysis:
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Fourier Analysis

An Introduction

Author: Elias M. Stein,Rami Shakarchi

Publisher: Princeton University Press

ISBN: 1400831237

Category: Mathematics

Page: 328

View: 5832

This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when studying problems in the physical sciences--that an arbitrary function can be written as an infinite sum of the most basic trigonometric functions. The first part implements this idea in terms of notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform; a clear introduction to the subject serves to avoid technical difficulties. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression. In organizing their exposition, the authors have carefully balanced an emphasis on key conceptual insights against the need to provide the technical underpinnings of rigorous analysis. Students of mathematics, physics, engineering and other sciences will find the theory and applications covered in this volume to be of real interest. The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Fourier Analysis is the first, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.
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Functional Analysis

Introduction to Further Topics in Analysis

Author: Elias M. Stein,Rami Shakarchi

Publisher: Princeton University Press

ISBN: 0691113874

Category: Mathematics

Page: 423

View: 6376

"This book covers such topics as Lp̂ spaces, distributions, Baire category, probability theory and Brownian motion, several complex variables and oscillatory integrals in Fourier analysis. The authors focus on key results in each area, highlighting their importance and the organic unity of the subject"--Provided by publisher.
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Functional Analysis

Author: P. K. Jain,Khalil Ahmad,Om P. Ahuja

Publisher: New Age International

ISBN: 9788122408010

Category: Functional analysis

Page: 326

View: 9302

The Book Is Intended To Serve As A Textbook For An Introductory Course In Functional Analysis For The Senior Undergraduate And Graduate Students. It Can Also Be Useful For The Senior Students Of Applied Mathematics, Statistics, Operations Research, Engineering And Theoretical Physics. The Text Starts With A Chapter On Preliminaries Discussing Basic Concepts And Results Which Would Be Taken For Granted Later In The Book. This Is Followed By Chapters On Normed And Banach Spaces, Bounded Linear Operators, Bounded Linear Functionals. The Concept And Specific Geometry Of Hilbert Spaces, Functionals And Operators On Hilbert Spaces And Introduction To Spectral Theory. An Appendix Has Been Given On Schauder Bases.The Salient Features Of The Book Are: * Presentation Of The Subject In A Natural Way * Description Of The Concepts With Justification * Clear And Precise Exposition Avoiding Pendantry * Various Examples And Counter Examples * Graded Problems Throughout Each ChapterNotes And Remarks Within The Text Enhances The Utility Of The Book For The Students.
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Problems and Solutions for Complex Analysis

Author: Rami Shakarchi

Publisher: Springer Science & Business Media

ISBN: 146121534X

Category: Mathematics

Page: 246

View: 4407

All the exercises plus their solutions for Serge Lang's fourth edition of "Complex Analysis," ISBN 0-387-98592-1. The problems in the first 8 chapters are suitable for an introductory course at undergraduate level and cover power series, Cauchy's theorem, Laurent series, singularities and meromorphic functions, the calculus of residues, conformal mappings, and harmonic functions. The material in the remaining 8 chapters is more advanced, with problems on Schwartz reflection, analytic continuation, Jensen's formula, the Phragmen-Lindeloef theorem, entire functions, Weierstrass products and meromorphic functions, the Gamma function and Zeta function. Also beneficial for anyone interested in learning complex analysis.
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Singular Integrals and Differentiability Properties of Functions (PMS-30)

Author: Elias M. Stein

Publisher: Princeton University Press

ISBN: 1400883881

Category: Mathematics

Page: 304

View: 4164

Singular integrals are among the most interesting and important objects of study in analysis, one of the three main branches of mathematics. They deal with real and complex numbers and their functions. In this book, Princeton professor Elias Stein, a leading mathematical innovator as well as a gifted expositor, produced what has been called the most influential mathematics text in the last thirty-five years. One reason for its success as a text is its almost legendary presentation: Stein takes arcane material, previously understood only by specialists, and makes it accessible even to beginning graduate students. Readers have reflected that when you read this book, not only do you see that the greats of the past have done exciting work, but you also feel inspired that you can master the subject and contribute to it yourself. Singular integrals were known to only a few specialists when Stein's book was first published. Over time, however, the book has inspired a whole generation of researchers to apply its methods to a broad range of problems in many disciplines, including engineering, biology, and finance. Stein has received numerous awards for his research, including the Wolf Prize of Israel, the Steele Prize, and the National Medal of Science. He has published eight books with Princeton, including Real Analysis in 2005.
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Visual Complex Analysis

Author: Tristan Needham

Publisher: Oxford University Press

ISBN: 9780198534464

Category: Mathematics

Page: 592

View: 9669

Now available in paperback, this successful radical approach to complex analysis replaces the standard calculational arguments with new geometric ones. With several hundred diagrams, and far fewer prerequisites than usual, this is the first visual intuitive introduction to complex analysis. Although designed for use by undergraduates in mathematics and science, the novelty of the approach will also interest professional mathematicians.
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Invitation to Complex Analysis

Author: Ralph P. Boas

Publisher: MAA

ISBN: 0883857642

Category: Mathematics

Page: 327

View: 2117

An ideal choice for a first course in complex analysis, this book can be used either as a classroom text or for independent study. Written in an informal style by a master expositor, the book distills more than half a century of experience with the subject into a lucid, engaging, yet rigorous account. The book reveals both the power of complex analysis as a tool for applications and the intrinsic beauty of the subject as a fundamental part of pure mathematics. Written at the level of courses commonly taught in American universities to seniors and beginning graduate students, the book is suitable for readers acquainted with advanced calculus or introductory real analysis. The treatment goes beyond the standard material of power series, Cauchy's theorem, residues, conformal mapping, and harmonic functions by including accessible discussions of many intriguing topics that are uncommon in a book at this level. Readers will encounter notions ranging from Landau's notation to overconvergent series to the Phragmén-Lindelöf theorem. The flexibility afforded by the supplementary topics and applications makes the book adaptable either to a short, one-term course or to a comprehensive, full-year course.The writing is user-friendly in many ways. Each topic is discussed in a typical, commonly encountered situation rather than in the most general, abstract setting. There are no numbered equations. Numerous exercises interspersed in the text encourage readers to test their understanding of new concepts and techniques as they are presented. Detailed solutions of the exercises, included at the back of the book, both serve as models for students and facilitate independent study. Supplementary exercises at the ends of sections, not solved in the book, provide an additional teaching tool.This second edition of Invitation to Complex Analysis has been painstakingly revised by the author's son, himself an award-winning mathematical expositor.
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Essays on Fourier Analysis in Honor of Elias M. Stein (PMS-42)

Author: Charles Fefferman,Robert Fefferman,Stephen Wainger

Publisher: Princeton University Press

ISBN: 1400852943

Category: Mathematics

Page: 396

View: 591

This book contains the lectures presented at a conference held at Princeton University in May 1991 in honor of Elias M. Stein's sixtieth birthday. The lectures deal with Fourier analysis and its applications. The contributors to the volume are W. Beckner, A. Boggess, J. Bourgain, A. Carbery, M. Christ, R. R. Coifman, S. Dobyinsky, C. Fefferman, R. Fefferman, Y. Han, D. Jerison, P. W. Jones, C. Kenig, Y. Meyer, A. Nagel, D. H. Phong, J. Vance, S. Wainger, D. Watson, G. Weiss, V. Wickerhauser, and T. H. Wolff. The topics of the lectures are: conformally invariant inequalities, oscillatory integrals, analytic hypoellipticity, wavelets, the work of E. M. Stein, elliptic non-smooth PDE, nodal sets of eigenfunctions, removable sets for Sobolev spaces in the plane, nonlinear dispersive equations, bilinear operators and renormalization, holomorphic functions on wedges, singular Radon and related transforms, Hilbert transforms and maximal functions on curves, Besov and related function spaces on spaces of homogeneous type, and counterexamples with harmonic gradients in Euclidean space. Originally published in 1995. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
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Complex Analysis in one Variable

Author: NARASIMHAN

Publisher: Springer Science & Business Media

ISBN: 1475711069

Category: Mathematics

Page: 268

View: 2951

This book is based on a first-year graduate course I gave three times at the University of Chicago. As it was addressed to graduate students who intended to specialize in mathematics, I tried to put the classical theory of functions of a complex variable in context, presenting proofs and points of view which relate the subject to other branches of mathematics. Complex analysis in one variable is ideally suited to this attempt. Of course, the branches of mathema tics one chooses, and the connections one makes, must depend on personal taste and knowledge. My own leaning towards several complex variables will be apparent, especially in the notes at the end of the different chapters. The first three chapters deal largely with classical material which is avai lable in the many books on the subject. I have tried to present this material as efficiently as I could, and, even here, to show the relationship with other branches of mathematics. Chapter 4 contains a proof of Picard's theorem; the method of proof I have chosen has far-reaching generalizations in several complex variables and in differential geometry. The next two chapters deal with the Runge approximation theorem and its many applications. The presentation here has been strongly influenced by work on several complex variables.
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Introduction to Complex Analysis

Author: H. A. Priestley

Publisher: OUP Oxford

ISBN: 0191037206

Category: Mathematics

Page: 344

View: 1652

Complex analysis is a classic and central area of mathematics, which is studied and exploited in a range of important fields, from number theory to engineering. Introduction to Complex Analysis was first published in 1985, and for this much awaited second edition the text has been considerably expanded, while retaining the style of the original. More detailed presentation is given of elementary topics, to reflect the knowledge base of current students. Exercise sets have been substantially revised and enlarged, with carefully graded exercises at the end of each chapter. This is the latest addition to the growing list of Oxford undergraduate textbooks in mathematics, which includes: Biggs: Discrete Mathematics 2nd Edition, Cameron: Introduction to Algebra, Needham: Visual Complex Analysis, Kaye and Wilson: Linear Algebra, Acheson: Elementary Fluid Dynamics, Jordan and Smith: Nonlinear Ordinary Differential Equations, Smith: Numerical Solution of Partial Differential Equations, Wilson: Graphs, Colourings and the Four-Colour Theorem, Bishop: Neural Networks for Pattern Recognition, Gelman and Nolan: Teaching Statistics.
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Introduction to Fourier Analysis on Euclidean Spaces (PMS-32)

Author: Elias M. Stein,Guido Weiss

Publisher: Princeton University Press

ISBN: 140088389X

Category: Mathematics

Page: 312

View: 6205

The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations, and to motivate the study of harmonic analysis on more general spaces having an analogous structure, e.g., symmetric spaces.
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Analytic Combinatorics

Author: Philippe Flajolet,Robert Sedgewick

Publisher: Cambridge University Press

ISBN: 1139477161

Category: Mathematics

Page: N.A

View: 4482

Analytic combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the analysis of algorithms and for the study of scientific models in many disciplines, including probability theory, statistical physics, computational biology, and information theory. With a careful combination of symbolic enumeration methods and complex analysis, drawing heavily on generating functions, results of sweeping generality emerge that can be applied in particular to fundamental structures such as permutations, sequences, strings, walks, paths, trees, graphs and maps. This account is the definitive treatment of the topic. The authors give full coverage of the underlying mathematics and a thorough treatment of both classical and modern applications of the theory. The text is complemented with exercises, examples, appendices and notes to aid understanding. The book can be used for an advanced undergraduate or a graduate course, or for self-study.
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Introductory Lectures on Automorphic Forms

Author: Walter L. Baily Jr.

Publisher: Princeton University Press

ISBN: 1400867150

Category: Mathematics

Page: 280

View: 7555

Intended as an introductory guide, this work takes for its subject complex, analytic, automorphic forms and functions on (a domain equivalent to) a bounded domain in a finite-dimensional, complex, vector space, usually denoted Cn). Part I, essentially elementary, deals with complex analytic automorphic forms on a bounded domain; it presents H. Cartan's proof of the existence of the projective imbedding of the compact quotient of such a domain by a discrete group. Part II treats the construction and properties of automorphic forms with respect to an arithmetic group acting on a bounded symmetric domain; this part is highly technical, and based largely on relevant results in functional analysis due to Godement and Harish-Chandra. In Part III, Professor Baily extends the discussion to include some special topics, specifically, the arithmetic propertics of Eisenstein series and their connection with the arithmetic theory of quadratic forms. Unlike classical works on the subject, this book deals with more than one variable, and it differs notably in its treatment of analysis on the group of automorphisms of the domain. It is concerned with the case of complex analytic automorphic forms because of their connection with algebraic geometry, and so is distinct from other modern treatises that deal with automorphic forms on a semi-simple Lie group. Having had its inception as graduate- level lectures, the book assumes some knowledge of complex function theory and algebra, for the serious reader is expected to supply certain details for himself, especially in such related areas as functional analysis and algebraic groups. Originally published in 1973. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
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From Stein to Weinstein and Back

Symplectic Geometry of Affine Complex Manifolds

Author: Kai Cieliebak,Y. Eliashberg

Publisher: American Mathematical Soc.

ISBN: 0821885332

Category: Mathematics

Page: 364

View: 9356

A beautiful and comprehensive introduction to this important field. --Dusa McDuff, Barnard College, Columbia University This excellent book gives a detailed, clear, and wonderfully written treatment of the interplay between the world of Stein manifolds and the more topological and flexible world of Weinstein manifolds. Devoted to this subject with a long history, the book serves as a super introduction to this area and also contains the authors' new results. --Tomasz Mrowka, MIT This book is devoted to the interplay between complex and symplectic geometry in affine complex manifolds. Affine complex (a.k.a. Stein) manifolds have canonically built into them symplectic geometry which is responsible for many phenomena in complex geometry and analysis. The goal of the book is the exploration of this symplectic geometry (the road from ``Stein to Weinstein'') and its applications in the complex geometric world of Stein manifolds (the road ``back''). This is the first book which systematically explores this connection, thus providing a new approach to the classical subject of Stein manifolds. It also contains the first detailed investigation of Weinstein manifolds, the symplectic counterparts of Stein manifolds, which play an important role in symplectic and contact topology. Assuming only a general background from differential topology, the book provides introductions to the various techniques from the theory of functions of several complex variables, symplectic geometry, $h$-principles, and Morse theory that enter the proofs of the main results. The main results of the book are original results of the authors, and several of these results appear here for the first time. The book will be beneficial for all students and mathematicians interested in geometric aspects of complex analysis, symplectic and contact topology, and the interconnections between these subjects.|This book is devoted to the interplay between complex and symplectic geometry in affine complex manifolds. Affine complex (a.k.a. Stein) manifolds have canonically built into them symplectic geometry which is responsible for many phenomena in complex geometry and analysis. The goal of the book is the exploration of this symplectic geometry (the road from ""Stein to Weinstein"") and its applications in the complex geometric world of Stein manifolds (the road ""back""). This is the first book which systematically explores this connection, thus providing a new approach to the classical subject of Stein manifolds. It also contains the first detailed investigation of Weinstein manifolds, the symplectic counterparts of Stein manifolds, which play an important role in symplectic and contact topology.

Assuming only a general background from differential topology, the book provides introductions to the various techniques from the theory of functions of several complex variables, symplectic geometry, $h$-principles, and Morse theory that enter the proofs of the main results. The main results of the book are original results of the authors, and several of these results appear here for the first time. The book will be beneficial for all students and mathematicians interested in geometric aspects of complex analysis, symplectic and contact topology, and the interconnections between these subjects.
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Harmonic Analysis (PMS-43)

Real-Variable Methods, Orthogonality, and Oscillatory Integrals. (PMS-43)

Author: Elias M. Stein

Publisher: Princeton University Press

ISBN: 140088392X

Category: Mathematics

Page: 712

View: 4467

This book contains an exposition of some of the main developments of the last twenty years in the following areas of harmonic analysis: singular integral and pseudo-differential operators, the theory of Hardy spaces, L\sup\ estimates involving oscillatory integrals and Fourier integral operators, relations of curvature to maximal inequalities, and connections with analysis on the Heisenberg group.
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An Introduction to Analysis

Author: Robert C. Gunning

Publisher: Princeton University Press

ISBN: 1400889413

Category: Mathematics

Page: 384

View: 3027

An essential undergraduate textbook on algebra, topology, and calculus An Introduction to Analysis is an essential primer on basic results in algebra, topology, and calculus for undergraduate students considering advanced degrees in mathematics. Ideal for use in a one-year course, this unique textbook also introduces students to rigorous proofs and formal mathematical writing--skills they need to excel. With a range of problems throughout, An Introduction to Analysis treats n-dimensional calculus from the beginning—differentiation, the Riemann integral, series, and differential forms and Stokes's theorem—enabling students who are serious about mathematics to progress quickly to more challenging topics. The book discusses basic material on point set topology, such as normed and metric spaces, topological spaces, compact sets, and the Baire category theorem. It covers linear algebra as well, including vector spaces, linear mappings, Jordan normal form, bilinear mappings, and normal mappings. Proven in the classroom, An Introduction to Analysis is the first textbook to bring these topics together in one easy-to-use and comprehensive volume. Provides a rigorous introduction to calculus in one and several variables Introduces students to basic topology Covers topics in linear algebra, including matrices, determinants, Jordan normal form, and bilinear and normal mappings Discusses differential forms and Stokes's theorem in n dimensions Also covers the Riemann integral, integrability, improper integrals, and series expansions
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Investors and Markets

Portfolio Choices, Asset Prices, and Investment Advice

Author: William F. Sharpe

Publisher: Princeton University Press

ISBN: 1400830184

Category: Business & Economics

Page: 232

View: 923

In Investors and Markets, Nobel Prize-winning financial economist William Sharpe shows that investment professionals cannot make good portfolio choices unless they understand the determinants of asset prices. But until now asset-price analysis has largely been inaccessible to everyone except PhDs in financial economics. In this book, Sharpe changes that by setting out his state-of-the-art approach to asset pricing in a nonmathematical form that will be comprehensible to a broad range of investment professionals, including investment advisors, money managers, and financial analysts. Bridging the gap between the best financial theory and investment practice, Investors and Markets will help investment professionals make better portfolio choices by being smarter about asset prices. Based on Sharpe's Princeton Lectures in Finance, Investors and Markets presents a method of analyzing asset prices that accounts for the real behavior of investors. Sharpe makes this technique accessible through a new, one-of-a-kind computer program (available for free on his Web site, at http://www.stanford.edu/~wfsharpe/apsim/index.html) that enables users to create virtual markets, setting the starting conditions and then allowing trading until equilibrium is reached and trading stops. Program users can then analyze the final portfolios and asset prices, see expected returns, and measure risk. In addition to popularizing the most sophisticated form of asset-price analysis, Investors and Markets summarizes much of Sharpe's most important previous work and reflects a lifetime of thinking about investing by one of the leading minds in financial economics. Any serious investment professional will benefit from Sharpe's unique insights.
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