Complex analysis, complex variables Books

Book SynopsisIntended for the students of mathematics, physics, engineering and other sciences, this second volume, in a three-part introduction to the subject, introduces the core areas of mathematical analysis while also illustrating the organic unity between them. It includes numerous examples and applications.Trade Review"Elias M. Stein, Winner of the 2005 Stefan Bergman Prize, American Mathematical Society"Table of ContentsForeword vii Introduction xv Chapter 1. Preliminaries to Complex Analysis 1 1 Complex numbers and the complex plane 1 1.1 Basic properties 1 1.2 Convergence 5 1.3 Sets in the complex plane 5 2 Functions on the complex plane 8 2.1 Continuous functions 8 2.2 Holomorphic functions 8 2.3 Power series 14 3 Integration along curves 18 4 Exercises 24 Chapter 2. Cauchy's Theorem and Its Applications 32 1 Goursat's theorem 34 2 Local existence of primitives and Cauchy's theorem in a disc 37 3 Evaluation of some integrals 41 4 Cauchy's integral formulas 45 5 Further applications 53 5.1 Morera's theorem 53 5.2 Sequences of holomorphic functions 53 5.3 Holomorphic functions defined in terms of integrals 55 5.4 Schwarz reflection principle 57 5.5 Runge's approximation theorem 60 6 Exercises 64 7 Problems 67 Chapter 3. Meromorphic Functions and the Logarithm 71 1 Zeros and poles 72 2 The residue formula 76 2.1 Examples 77 3 Singularities and meromorphic functions 83 4 The argument principle and applications 89 5 Homotopies and simply connected domains 93 6 The complex logarithm 97 7 Fourier series and harmonic functions 101 8 Exercises 103 9 Problems 108 Chapter 4. The Fourier Transform 111 1 The class F 113 2 Action of the Fourier transform on F 114 3 Paley-Wiener theorem 121 4 Exercises 127 5 Problems 131 Chapter 5. Entire Functions 134 1 Jensen's formula 135 2 Functions of finite order 138 3 Infinite products 140 3.1 Generalities 140 3.2 Example: the product formula for the sine function 142 4 Weierstrass infinite products 145 5 Hadamard's factorization theorem 147 6 Exercises 153 7 Problems 156 Chapter 6. The Gamma and Zeta Functions 159 1 The gamma function 160 1.1 Analytic continuation 161 1.2 Further properties of T 163 2 The zeta function 168 2.1 Functional equation and analytic continuation 168 3 Exercises 174 4 Problems 179 Chapter 7. The Zeta Function and Prime Number Theorem 181 1 Zeros of the zeta function 182 1.1 Estimates for 1/s(s) 187 2 Reduction to the functions v and v1 188 2.1 Proof of the asymptotics for v1 194 Note on interchanging double sums 197 3 Exercises 199 4 Problems 203 Chapter 8. Conformal Mappings 205 1 Conformal equivalence and examples 206 1.1 The disc and upper half-plane 208 1.2 Further examples 209 1.3 The Dirichlet problem in a strip 212 2 The Schwarz lemma; automorphisms of the disc and upper half-plane 218 2.1 Automorphisms of the disc 219 2.2 Automorphisms of the upper half-plane 221 3 The Riemann mapping theorem 224 3.1 Necessary conditions and statement of the theorem 224 3.2 Montel's theorem 225 3.3 Proof of the Riemann mapping theorem 228 4 Conformal mappings onto polygons 231 4.1 Some examples 231 4.2 The Schwarz-Christoffel integral 235 4.3 Boundary behavior 238 4.4 The mapping formula 241 4.5 Return to elliptic integrals 245 5 Exercises 248 6 Problems 254 Chapter 9. An Introduction to Elliptic Functions 261 1 Elliptic functions 262 1.1 Liouville's theorems 264 1.2 The Weierstrass p function 266 2 The modular character of elliptic functions and Eisenstein series 273 2.1 Eisenstein series 273 2.2 Eisenstein series and divisor functions 276 3 Exercises 278 4 Problems 281 Chapter 10. Applications of Theta Functions 283 1 Product formula for the Jacobi theta function 284 1.1 Further transformation laws 289 2 Generating functions 293 3 The theorems about sums of squares 296 3.1 The two-squares theorem 297 3.2 The four-squares theorem 304 4 Exercises 309 5 Problems 314 Appendix A: Asymptotics 318 1 Bessel functions 319 2 Laplace's method; Stirling's formula 323 3 The Airy function 328 4 The partition function 334 5 Problems 341 Appendix B: Simple Connectivity and Jordan Curve Theorem 344 1 Equivalent formulations of simple connectivity 345 2 The Jordan curve theorem 351 2.1 Proof of a general form of Cauchy's theorem 361 Notes and References 365 Bibliography 369 Symbol Glossary 373 Index 375

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Book SynopsisTrade Review“A clear analysis of systems ranging from radiation to human brains”—Nature“Many people might not bother to define complexity, thinking that we know it when we see it. Scientists and philosophers have no such luxury, and for them this book will be invaluable. Ladyman and Wiesner have provided a compact but comprehensive overview of the different ways that systems can be complex, ultimately arguing that complexity comes in distinct forms, but that their commonalities are nevertheless quite real.”—Sean Carroll, author of Something Deeply Hidden: Quantum Worlds and the Emergence of Spacetime“This is an outstanding, original, and much-needed book. Ladyman and Wiesner give an accessible, engaging, and precise overview of complexity science from a panoptic perspective, spanning many different kinds of examples from a variety of disciplines”—James Owen Weatherall, coauthor of The Misinformation Age: How False Beliefs Spread“Written in a lively and readable style, What Is a Complex System? provides a clear and coherent synthesis of the myriad and sometimes contradictory descriptions and definitions of complex systems.”—Colm Connaughton, Director of the Centre for Complexity Science, University of Warwick “This is highly thoughtful incisive essay on the meaning and use of the concept of complex systems. I particularly like the attempt to formulate syntheses across fields, across features and across mechanisms.”—Didier Sornette, author of Why Stock Markets Crash: Critical Events in Complex Financial Systems“This book is a superb introduction to complex systems. Ladyman and Wiesner skillfully guide the reader from examples of complex systems all-around us, to ten common features of such systems and how to mathematically measure them, to a discussion of complexity as a scientific field by itself. Anyone interested in complex systems should read this book before any other.”—Tina Eliassi-Rad, Professor of Network Science, Northeastern University

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Book SynopsisTimothy Sauer earned his Ph.D. in mathematics at the University of CaliforniaBerkeley in 1982, and is currently a professor at George Mason University. He has published articles on a wide range of topics in applied mathematics, including dynamical systems, computational mathematics, and mathematical biology.Table of Contents Preface 1. Solving Equations 2. Systems of Equations 3. Interpolation 4. Least Squares 5. Numerical Differentiation and Integration 6. Ordinary Differential Equations 7. Boundary Value Problems 8. Partial Differential Equations 9. Random Numbers and Applications 10. Trigonometric Interpolation and the FFT 11. Compression 12. Eigenvalues and Singular Values References

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Book SynopsisHelps you study the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This title introduces some key ideas in the field, and to form a basis for further study.Trade Review"John Milnor's book provides a solid foundation and the kind of bird's eye view that perhaps only a mathematician of his caliber can offer."--William J. Satzer, MAA ReviewsTable of Contents*FrontMatter, pg. i*Table Of Contents, pg. v*List of Figures, pg. vi*Preface to the Third Edition, pg. vii*Chronological Table, pg. viii*Riemann Surfaces, pg. 1*Iterated Holomorphic Maps, pg. 39*Local Fixed Point Theory, pg. 76*Periodic Points: Global Theory, pg. 142*Structure of the Fatou Set, pg. 161*Using the Fatou Set to Study the Julia Set, pg. 174*Appendix A. Theorems from Classical Analysis, pg. 219*Appendix B. Length-Area-Modulus Inequalities, pg. 226*Appendix C. Rotations, Continued Fractions, and Rational Approximation, pg. 234*Appendix D. Two or More Complex Variables, pg. 246*Appendix E. Branched Coverings and Orbifolds, pg. 254*Appendix F. No Wandering Fatou Components, pg. 259*Appendix G. Parameter Spaces, pg. 266*Appendix H. Computer Graphics and Effective Computation, pg. 271*References, pg. 277*Index, pg. 293

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Book SynopsisTrade Review"A no-nonsense, clearly written graduate level textbook . . . . far more approachable than many other books on complex analysis"---Jonathan Shock, Mathemafrica"An excellent textbook. . . . Carefully and precisely written in a lively style."---Ali Abkar, zbMATH Open"Beautifully produced, beautifully written, on an incomparably beautiful area of mathematics, this is an inspirational book that I shall gratefully return to again and again."---Nick Lord, Mathematical Gazette

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Book SynopsisThis accessible guide walks you through the math necessary to thrive in the AI field such as focusing on real-world applications rather than dense academic theory. Engineers, data scientists, and students alike will examine mathematical topics critical for AI-including regression, neural networks, optimization, backpropagation, and Markov chains.

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Book SynopsisAimed at graduate students, this textbook provides an accessible and comprehensive introduction to operator theory, and covers twenty examples of operators, discussing the norm, spectrum, commutant, invariant subspaces, and interesting properties of each operator.Trade ReviewThe text is supplemented by over 600 end-of-chapter exercises, designed to help the reader master the topics covered in the chapter, as well as providing an opportunity to further explore the vast operator theory literature. Each chapter also contains well researched historical facts which place each chapter within the broader context of the development of the field as a whole. * MathSciNet *Table of Contents1: Hilbert Spaces 2: Diagonal Operators 3: Infinite Matrices 4: Two Multiplication Operators 5: The Unilateral Shift 6: The Cesàro Operator 7: The Volterra Operator 8: Multiplication Operators 9: The Dirichlet Shift 10: The Bergman Shift 11: The Fourier Transform 12: The Hilbert Transform 13: Bishop Operators 14: Operator Matrices 15: Constructions with the Shift Operator 16: Toeplitz Operators 17: Hankel Operators 18: Composition Operators 19: Subnormal Operators 20: The Compressed Shift

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Book SynopsisThis monograph treats the analysis of symmetric cones in a systematic way. It discusses harmonic analysis and special functions associated with symmetric cones; it also tries these results together with the study of holomorphic functions on bounded symmetric domains of tube type.Trade Review... the present book is more carefully directed at the graduate student level, includes numerous exercises, and has its emphasis more on the harmonic analysis side. Such a presentation is much needed. The detailed exposition, careful choice of organization and notation, and very helpful collection of exercises, mostly of medium difficulty, all attest to the effort put into this joint venture. As a highly readable and accessible presentation of Jordan algebras and their applications to Riemannian geometry and harmonic analysis, the book is strongly recommended to all analysts (starting at graduate level) working in the multi-variable setting of symmetric spaces and Lie groups. Bulletin of the London Mathematical SocietyTable of ContentsI. Convex cones ; II. Jordan algebras ; III. Symmetric cones and Euclidean Jordan algebras ; IV. The Peirce decomposition in a Jordan algebra ; V. Classification of Euclidean Jordan algebras ; VI. Polar decomposition and Gauss decomposition ; VII. The gamma function of a symmetric cone ; VIII. Complex Jordan algebras ; IX. Tube domains over convex cones ; X. Symmetric domains ; XI. Conical and spherical polynomials ; XII. Taylor and Laurent series ; XIII. Functions spaces on symmetric domains ; XIV. Invariant differential operators and spherical functions ; XV. Special functions ; XVI. Representations of Jordan algebras and Euclidean Fourier analysis ; Bibliography

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Book SynopsisPresents easy to understand proofs of some of the most difficult results about polynomials demonstrated by means of applications.Trade ReviewPresents easy to understand proofs of some of the most difficult results about polynomials demonstrated by means of applications ... Brings to the subject an immense range of reference to the study of polynomials. Professional and academic mathematicians of complex analysis, approximation theory and theoretical numerical analysis; graduate students in mathematics; engineers, statisticians and theoretical physicists, who have an interest in the important results about polynomials, will not do better than start with reading and referring to this book. * Current Engineering Practice *A nicely written book that will be useful for scientists, engineers and mathematicians from other fields. It can be strongly recommended as an undergraduate or graduate text and as a comprehensive source for self study. * EMS *Table of Contents2. FUNDAMENTAL RESULTS ON CRITICAL POINTS ; 8. INCLUSION OF ALL ZEROS ; 12. GROWTH ESTIMATES

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Book SynopsisBased on a series of graduate lectures given by Vladimir Markovic at the University of Warwick in spring 2003, this book is accessible to those with a grounding in complex analysis looking for an introduction to the theory of quasiconformal maps and Teichmüller theory. Assuming some familiarity with Riemann surfaces and hyperbolic geometry, topics covered include the Grötzch argument, analytical properties of quasiconformal maps, the Beltrami differential equation, holomorphic motions and Teichmüller spaces. Where proofs are omitted, references to where they may be found are always given, and the text is clearly illustrated throughout with diagrams, examples, and exercises for the reader.Table of ContentsPreface ; 1. The Grotzch argument ; 2. Geometric definition of quasiconformal maps ; 3. Analytic properties of quasiconformal maps ; 4. Quasi-isometries and quasisymmetric maps ; 5. The Beltrami differential equation ; 6. Holomorphic motions and applications ; 7. Teichmuller spaces ; 8. Extremal quasiconformal mappings ; 9. Unique extremality ; 10. Isomorphisms of Teichmuller space ; 11. Local rigidity of Teichmuller spaces ; References ; Index

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Book SynopsisSystems biology came about as growing numbers of engineers and scientists from other fields created algorithms which supported the analysis of biological data in incredible quantities. Whereas biologists of the past had been forced to study one item or aspect at a time, due to technical and biological limitations, it suddenly became possible to study biological phenomena within their natural contexts. This interdisciplinary field offers a holistic approach to interpreting these processes, and has been responsible for some of the most important developments in the science of human health and environmental sustainability. This Very Short Introduction outlines the exciting processes and possibilities in the new field of systems biology. Eberhard O. Voit describes how it enabled us to learn how intricately the expression of every gene is controlled, how signaling systems keep organisms running smoothly, and how complicated even the simplest cells are. He explores what this field is about, why it is needed, and how it will affect our understanding of life, particularly in the areas of personalized medicine, drug development, food and energy production, and sustainable stewardship of our environments. Throughout he considers how new tools are being provided from the fields of mathematics, computer science, engineering, physics, and chemistry to grasp the complexity of the countless interacting processes in cells which would overwhelm the cognitive and analytical capabilities of the human mind. ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.Table of ContentsAcknowledgments 1: What is systems biology all about? 2: The exciting puzzles of computational systems biology 3: The -omics revolution 4: Computational systems biology (CSB) 5: Interdependencies of biological systems 6: Simulators 7: The lawless pursuit of biological systems References Further reading Index

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Book SynopsisBanach spaces and algebras are a key topic of pure mathematics. Graham Allan''s careful and detailed introductory account will prove essential reading for anyone wishing to specialise in functional analysis and is aimed at final year undergraduates or masters level students. Based on the author''s lectures to fourth year students at Cambridge University, the book assumes knowledge typical of first degrees in mathematics, including metric spaces, analytic topology, and complex analysis. However, readers are not expected to be familiar with the Lebesgue theory of measure and integration.The text begins by giving the basic theory of Banach spaces, including dual spaces and bounded linear operators. It establishes forms of the theorems that are the pillars of functional analysis, including the Banach-Alaoglu, Hahn-Banach, uniform boundedness, open mapping, and closed graph theorems. There are applications to Fourier series and operators on Hilbert spaces.The main body of the text is an intTrade ReviewThis well-crafted and scholarly book ...leaves nothing to be desired: this is a fine way to get into this beautiful in a subject and will serve to reel in a huge number of futureews devotees. * Michael Berg, MAA Reviews *Table of ContentsPART I INTRODUCTION TO BANACH SPACES ; 1. Preliminaries ; 2. Elements of normed spaces ; 3. Banach spaces ; PART II BANACH ALGEBRAS ; 4. Banach algebras ; 5. Representation theory ; 6. Algebras with an involution ; 7. The Borel functional calculus ; PART III SCV AND BANACH ALGEBRAS ; 8. Introduction to several complex variables ; 9. The holomorphic functional calculus in several variables ; Bibliography ; Index

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Book SynopsisThis textbook is designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors. The book has its origins in a series of lecture courses given by the authors, all of whom are internationally known mathematicians and renowned expositors. It is written in an accessible and informal style, and fills a gap in the existing literature. The introduction by Nigel Hitchin addresses the meaning of integrability: how do we recognize an integrable system? His own contribution then develops connections with algebraic geometry, and includes an introduction to Riemann surfaces, sheaves, and line bundles. Graeme Segal takes the Kortewegde Vries and nonlinear Schrödinger equations as central examples, and explores the mathematical structures underlying the inverse scattering transform. He explains the roles of loop groups, the Grassmannian, and algebraic curves. In the final part of the book, Richard Ward explores the connection beTrade ReviewThe subject of the book is fascinating and written versions of the lecture series are nicley presented and preserve well the informal spirit of the lectures. This is a very useful book for graduate students and for mathematicians (or physicists) from other fields interested in the topic. * EMS *The lecturers cover an enormous amount of material, ranging from algeraic geometry and the theory of Riemann surfaces to loop groups, connections, Yang-Mills equations and twister theory. However despite this wide range, the book is surprisingly self-contained and readable. * Bulletin of the London Mathematical Society *Table of Contents1. Introduction ; 2. Riemann surfaces and integrable systems ; 3. Integrable systems and inverse scattering ; 4. Integrable systems and twistors ; Index

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Book SynopsisThis up-to-date account brings together results previously scattered throughout the literature as well as new material in the area of function theory. The focus is on describing some of what has been learned thus far about the structure of the de Branges-Rovnyak spaces and their function-theoretic connections.Table of ContentsHilbert Spaces Inside Hilbert Spaces. Hilbert Spaces Inside H?. Cauchy Integral Representations. Nonextreme Points. Extreme Points. Angular Derivatives. Higher Derivatives. Equality of H(b) and H(). Equality of H(b) and M(a). Near Equality of H(b) and M(a). Brief Mention of a Few Additional Topics. References. Supplementary References. Index.

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Book SynopsisModern Real and Complex Analysis Thorough, well-written, and encyclopedic in its coverage, this text offers a lucid presentation of all the topics essential to graduate study in analysis. While maintaining the strictest standards of rigor, Professor Gelbaum's approach is designed to appeal to intuition whenever possible.Table of ContentsREAL ANALYSIS. Fundamentals. Integration. Functional Analysis. More Measure Theory. COMPLEX ANALYSIS. Locally Holomorphic Functions. Harmonic Functions. Meromorphic and Entire Functions. Conformal Mapping. Defective Functions. Riemann Surfaces. Convexity and Complex Analysis. Several Complex Variables. Bibliography. Symbol List. Glossary/Index.

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Book SynopsisSpatial data analysis is a fast growing area and Voronoi diagrams provide a means of naturally partitioning space into subregions to facilitate spatial data manipulation, modelling of spatial structures, pattern recognition and locational optimization.Trade Review"While this edition maintains the overall structure of the first, there are substantial changes in the content..." (Mathematical Reviews, Issue 2001c) "...a must..." (Monatshefte fur Mathematik, Vol 131/2, 2000)Table of ContentsDefinitions and Basic Properties of Voronoi Diagrams. Generalizations of the Voronoi Diagram. Algorithms for Computing Voronoi Diagrams. Poisson Voronoi Diagrams. Spatial Interpolation. Models of Spatial Processes. Point Pattern Analysis. Locational Optimization Through Voronoi Diagrams. References. Index.

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Book SynopsisDr Smithies analyses the process whereby Cauchy created one of the most important branch of mathematics discovered in the nineteenth century. It is the first book to make use of the whole spectrum of available original sources, and will be recognised as the authoritative work on the creation of complex function theory.Trade ReviewReview of the hardback: 'This exemplary book is the first thorough and comprehensive presentation of Cauchy's creation between 1814 and 1831, of complex function theory … the papers form a diary that testifies to the gradual development of a new field in the mind of an outstanding mathematician.' D. Laugwitz, DarmstadtReview of the hardback: ' … an invaluable guide to anyone interested in the work of a great mathematician'. Jeremy Gray, The Open UniversityReview of the hardback: 'This book is a welcome contribution to the history of mathematical ideas. It is carefully written … in the case of a great mathematician of the first half of the nineteenth century, it is very useful to have a book analyzing his ideas. Smithie's book helps us understand the technical achievements of Cauchy in founding the theory of complex functions.' The Mathematical GazetteReview of the hardback: '… new light is thrown on Cauchy's thinking.' L'Enseignment MathématiqueReview of the hardback: '… a unique source not only for the historians of mathematics but to all who are fascinated by the beauty of the complex function theory.' European Mathematical SocietyTable of Contents1. Introduction; 2. The background to Cauchy's work on complex function theory; 3. Cauchy's 1814 memoir on definite integrals; 4. Miscellaneous contributions (1815–1825); 5. The 1825 memoir and associated papers; 6. The calculus of residues; 7. The Lagrange series and the Turin memoirs; 8. Summary and conclusions; References.

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Book SynopsisProfessor Carathéodory sets out the basic theory of conformal representations as simply as possible. In the early chapters on Mobius' and other elementary transformations and on non-Euclidean geometry, he deals with those elementary subjects that are necessary for an understanding of the general theory discussed in the remaining chapters.Table of Contents1. Mobius Transformation; 2. Non-Euclidean Geometry; 3. Elementary Transformations; 4. Schwarz's Lemma; 5. The Fundamental Theorems of Conformal Representation; 6. Transformation of the Frontier; 7. Transformation of Closed Surfaces; 8. The General Theorem of Uniformisation.

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Book SynopsisIn this book the authors give elementary accounts of many aspects of classical complex function theory including Möbius transformations, elliptic functions, Riemann surfaces, Fuchsian groups and modular functions. A distinctive feature of their presentation is the way in which they have incorporated into the text many interesting topics from other branches of mathematics.Trade Review"...a very nice treatment which emphasizes the unity of mathematics...Several years ago the reviewer wanted to teach an undergraduate course that gave an introduction to hyperbolic geometry, Möbius transformations and discrete groups. There was no suitable undergraduate text. This book fills that void," Mathematical Reviews"Well motivated with good selection of problems at the end of each chapter." American Mathematical Monthly"...clear and well written...Its message, admirably conveyed, is that mathematics is not a collection of neat parcels, the contents of one being blind to the contents of another." Times Higher Education Supplement"...clearly presented...a delightful book..." American ScientistTable of ContentsIntroduction; 1. The Riemann sphere; 2. Möbius transformations; 3. Elliptic functions; 4. Meromorphic continuation and Riemann surfaces; 5. PSL(2,R) and its discrete subgroups; 6. The modular group; Appendices; References; Index of symbols; Index of names and definitions.

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Book SynopsisThe Riemann zeta function is one of the most studied objects in mathematics, and is of fundamental importance. In this book, based on his own research, Professor Motohashi shows that the function is closely bound with automorphic forms and that many results from there can be woven with techniques and ideas from analytic number theory to yield new insights into, and views of, the zeta function itself. The story starts with an elementary but unabridged treatment of the spectral resolution of the non-Euclidean Laplacian and the trace formulas. This is achieved by the use of standard tools from analysis rather than any heavy machinery, forging a substantial aid for beginners in spectral theory as well. These ideas are then utilized to unveil an image of the zeta-function, first perceived by the author, revealing it to be the main gem of a necklace composed of all automorphic L-functions. In this book, readers will find a detailed account of one of the most fascinating stories in the develoTrade ReviewReview of the hardback: '… gives an excellent presentation of the interplay between the Riemann zeta function and automorphic forms … nicely written and of great interest for any number theorists.' R. Tichy, International Mathematical NewsTable of Contents1. Non-Euclidean harmonics; 2. Trace formulas; 3. Automorphic L-functions; 4. An explicit formula; 5. Asymptotics; References; Index.

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Book SynopsisComplex variables provide powerful methods for attacking many difficult problems, and it is the aim of this book to provide a thorough grounding in these methods and their application. This 2003 edition was improved throughout and is ideal for use in undergraduate and introductory graduate courses in complex variables.Trade Review'… an excellent text, and one of the most complete and well-written books on complex variables I have seen … The index is nicely composed, complete, and accurate … useful as a reference … I highly recommend it to anyone interested in the subject and have placed it prominently upon my reference bookshelf.' Duwayne Anderson, Optics and Photonics News'… an excellent text, and one of the most complete and well-written books on complex variables I have seen … I highly recommend it to anyone interested in the subject…'. Optics and Photonics News'Overall the book feels 'road-tested' both in the lecture theatre and in the crucible of research and the clear well-written text is complemented by a superb collection of worked examples and exercises ranging in scope from routine applications of techniques to more substantial illustrations of the theory.' The Mathematical Gazette'… the book is valuable for students in engineering and physical sciences.' ZAMMTable of ContentsPart I: 1. Complex numbers and elementary functions; 2. Analytic functions and integration; 3. Sequences, series and singularities of complex functions; 4. Residue calculus and applications of contour integration; Part II: 5. Conformal mapping and applications; 6. Asymptotic evaluation of integrals; 7. Riemann–Hilbert problems; Index.

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Book SynopsisFew books on the subject of Riemann surfaces cover the relatively modern theory of dessins d'enfants (children's drawings), which was launched by Grothendieck in the 1980s and is now an active field of research. In this 2011 book, the authors begin with an elementary account of the theory of compact Riemann surfaces viewed as algebraic curves and as quotients of the hyperbolic plane by the action of Fuchsian groups of finite type. They then use this knowledge to introduce the reader to the theory of dessins d'enfants and its connection with algebraic curves defined over number fields. A large number of worked examples are provided to aid understanding, so no experience beyond the undergraduate level is required. Readers without any previous knowledge of the field of dessins d'enfants are taken rapidly to the forefront of current research.Trade Review"Overall the text is very well written and easy to follow, partly due to the abundance of good concrete examples in every single section illustrating concepts from the very basic to the very technical." Aaron D. Wootton, Mathematical ReviewsTable of Contents1. Riemann surfaces and algebraic curves; 2. Riemann surfaces and Fuchsian groups; 3. Belyi's theorem; 4. Dessins d'enfants; References; Index.

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Book SynopsisThis 2006 treatment of Jacobi elliptic functions seeks to answer the fascinating question: 'what would the treatment of elliptic functions have been like if Abel had developed the ideas, rather than Jacobi?' A rich variety of applications of the elliptic and related functions are also presented.Trade Review'This solid text is a good place to start when working with elliptic functions and it is the sort of book that you will keep coming back to as reference text.' Mathematics TodayTable of Contents1. The 'simple' pendulum; 2. Jacobian elliptic functions of a complex variable; 3. General properties of elliptic functions; 4. Theta functions; 5. The Jacobian elliptic functions for complex k; 6. Introduction to transformation theory; 7. The Weierstrass elliptic functions; 8. Elliptic integrals; 9. Applications of elliptic functions in geometry; 10. An application of elliptic functions in algebra solution of the general quintic equation; 11. An arithmetic application of elliptic functions; 12. Applications in mechanics and statistics and other topics; Appendix; Bibliography.

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Book SynopsisWritten by a master of the subject, this text will be appreciated by students and experts for the way it develops the classical theory of functions of a complex variable in a clear and straightforward manner. In general, the approach taken here emphasises geometrical aspects of the theory in order to avoid some of the topological pitfalls associated with this subject. Thus, Cauchy's integral formula is first proved in a topologically simple case from which the author deduces the basic properties of holomorphic functions. Starting from the basics, students are led on to the study of conformal mappings, Riemann's mapping theorem, analytic functions on a Riemann surface, and ultimately the RiemannâRoch and Abel theorems. Profusely illustrated, and with plenty of examples, and problems (solutions to many of which are included), this book should be a stimulating text for advanced courses in complex analysis.Trade Review'While most of the material included in the first part could be used in a basic course on complex analysis, the whole book could serve as a text for an advanced course on Riemann surfaces. The book contains many pictures (helping to build geometric intuition) and problems (elementary and advanced). The book could be very helpful for students as well as for experts in the field.' European Mathematical Society NewsletterTable of Contents1. Holomorphic functions; 2. Cauchy's theorem; 3. Conformal mappings; 4. Analytic continuation; 5. Riemann's mapping theorem; 6. Riemann surfaces; 7. The structure of Riemann surfaces; 8. Analytic functions on a closed Riemann surface.

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Book SynopsisL-functions associated to automorphic forms encode all classical number theoretic information. They are akin to elementary particles in physics. This book provides an entirely self-contained introduction to the theory of L-functions in a style accessible to graduate students with a basic knowledge of classical analysis, complex variable theory, and algebra. Also within the volume are many new results not yet found in the literature. The exposition provides complete detailed proofs of results in an easy-to-read format using many examples and without the need to know and remember many complex definitions. The main themes of the book are first worked out for GL(2,R) and GL(3,R), and then for the general case of GL(n,R). In an appendix to the book, a set of Mathematica functions is presented, designed to allow the reader to explore the theory from a computational point of view.Trade Review'… a gentle introduction to this fascinating new subject. The presentation is very explicit and many examples are worked out with great detail … This book should be of great interest to students beginning with the theory of modular forms or for more advanced readers wanting to know about general L-functions.' Emmanuel P. Royer, Mathematical Reviews'This book, whose clear and sometimes simplified proofs make the basic theory of automorphic forms on GL(n) accessible to a wide audience, will be valuable for students. It nicely complements D. Bump's book (Automorphic Forms and Representations, Cambridge, 1997), which offers a greater emphasis on representation theory and a different selection of topics.' Zentralblatt MATH'Unfortunately, when n > 2 the GL(n) theory is not very accessible to the student of analytic number theory, yet it is increasing in importance. [This book] addresses this problem by developing a large part of the theory in a way that is carefully designed to make the field accessible … much of the literature is written in the adele language, and seeing how it translates into classical terms is both useful and enlightening … This is a unique and very welcome book, one that the student of automorphic forms will want to study, and also useful to experts.' Daniel Bump, SIAM ReviewTable of ContentsIntroduction; 1. Discrete group actions; 2. Invariant differential operators; 3. Automorphic forms and L-functions for SL(2,Z); 4. Existence of Maass forms; 5. Maass forms and Whittaker functions for SL(n,Z); 6. Automorphic forms and L-functions for SL(3,Z); 7. The Gelbert–Jacquet lift; 8. Bounds for L-functions and Siegel zeros; 9. The Godement–Jacquet L-function; 10. Langlands Eisenstein series; 11. Poincaré series and Kloosterman sums; 12. Rankin–Selberg convolutions; 13. Langlands conjectures; Appendix. The GL(n)pack manual; References.

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Book SynopsisPresents many of the main developments in the study of real submanifolds in complex space, providing background material for researchers and advanced graduate students. This work addresses topics such as the holomorphic extension of functions and mappings that satisfy the tangential Cauchy-Riemann equations on real submanifolds.Table of ContentsPrefaceCh. IHypersurfaces and Generic Submanifolds in C[superscript N]3Ch. IIAbstract and Embedded CR Structures35Ch. IIIVector Fields: Commutators, Orbits, and Homogeneity62Ch. IVCoordinates for Generic Submanifolds94Ch. VRings of Power Series and Polynomial Equations119Ch. VIGeometry of Analytic Discs156Ch. VIIBoundary Values of Holomorphic Functions in Wedges184Ch. VIIIHolomorphic Extension of CR Functions205Ch. IXHolomorphic Extension of Mappings of Hypersurfaces241Ch. XSegre Sets281Ch. XINondegeneracy Conditions for Manifolds315Ch. XIIHolomorphic Mappings of Submanifolds349Ch. XIIIMappings of Real-algebraic Subvarieties379References390Index401

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Book SynopsisThe author introduced the concept of a "local system" on P1-{a finite set of points} nearly 140 years ago. His idea was to study nth order linear differential equations by studying the rank n local systems (of local holomorphic solutions) to which they gave rise.Trade Review"It is clear that this book presents highly important new views and results on the classical theory of complex linear differential equations."--Zentralblatt fur MathematikTable of Contents* First results on rigid local systems * The theory of middle concolution * Fourier Transform and rigidity * Middle concolution: dependence on parameters * Structure of rigid local systems * Existence algorithms for rigids * Diophantine aspects of rigidity * rigids

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Book SynopsisSpin glasses are disordered magnetic systems that have led to the development of mathematical tools with an array of real-world applications, from airline scheduling to neural networks. This book offers an introduction to the subject, explaining what spin glasses are, and how they are opening up new ways of thinking about complexity.Trade Review"The challenge that Stein and Newman faced in creating this book ... was to write for a broad range of readers and still offer interesting depth. As they state in the preface, they are aiming for a reading level that is between Scientific American and research journals. This reviewer believes they have succeeded... Stein and Newman write well and keep the mathematics to a minimum."--Choice "[A] surprisingly broad field of view is visible through the lens of the classical, equilibrium using spin glass and the authors are able to use it to explore many fascinating topics. Stein and Newman have written an excellent introduction to the field of spin glasses and the many ramifications of spin glass theory outside of condensed matter physics and statistical mechanics. Experts and novices alike will find this book interesting and useful."--Jonathan Machta, Journal of Statistical Physics "Spin Glasses and Complexity is not a journalistic book that merely reports on the subject. Based on profound mathematical insights, here distilled into an incisive presentation, it represents the fruit of the lifelong commitments two experts have made to spin-glass theory within and beyond physics... Spin Glasses and Complexity is unique in successfully bringing this thrilling theme to a broader scientific audience."--Stefan Boettcher, Physics Today "[T]he work is well presented and the reader will surely find it both inspiring and interesting."--Marco Castrillon Lopez, European Mathematical Society "Well presented and the reader will surely find it both inspiring and interesting."--Marco Castrillon Lopez, European Mathematical SocietyTable of ContentsPreface xi Introduction: Why Spin Glasses? 1 *1. Order, Symmetry, and the Organization of Matter 15 *1.1 The Symmetry of Physical Laws 17 *1.2 The Hamiltonian 23 *1.3 Broken Symmetry 26 *1.4 The Order Parameter 31 *1.5 Phases of Matter 35 *1.6 Phase Transitions 39 *1.7 Summary: The Unity of Condensed Matter Physics 41 2. Glasses and Quenchied Disorder 43 *2.1 Equilibrium and Non Equilibrium 43 * 2.2 The Glass Transition 45 *2.3 Localization 49 3. Magnetic Systems 51 *3.1 Spin 51 *3.2 Magnetism in Solids 53 *3.3 The Paramagnetic Phase 55 *3.4 Magnetization 55 *3.5 The Ferromagnetic Phase and Magnetic Susceptibility 57 *3.6 The Antiferromagnetic Phase 59 *3.7 Broken Symmetry and the Heisenberg Hamiltonian 59 4. Spin Glasses: General Features 63 *4.1 Dilute Magnetic Alloys and the Kondo Effect 64 *4.2 A New State of Matter? 65 *4.3 Nonequilibrium and Dynamical Behavior 71 *4.4 Mechanisms Underlying Spin Glass Behavior 74 *4.5 The Edwards-Anderson Hamiltonian 78 *4.6 Frustration 81 *4.7 Dimensionality and Phase Transitions 83 *4.8 Broken Symmetry and the Edwards-Anderson Order Parameter 85 *4.9 Energy Landscapes and Metastability 86 5. The Infinite-Range Spin Glass 90 *5.1 Mean Field Theory 90 *5.2 The Sherrington-Kirkpatrick Hamiltonian 92 *5.3 A Problem Arises 93 *5.4 The Remedy 95 *5.5 Thermodynamic States 97 *5.6 The Meaning of Replica Symmetry Breaking 98 *5.7 The Big Picture 109 6. Applications to Other Fields 112 *6.1 Computational Time Complexity and Combinatorial Optimization 113 *6.2 Neural Networks and Neural Computation 129 *6.3 Protein Folding and Conformational Dynamics 144 *6.4 Short Takes 168 7. Short-Range Spin Glasses: Some Basic Questions 175 *7.1 Ground States 177 *7.2 Pure States 188 *7.3 Scenarios for the Spin Glass Phase of the EA Model 193 *7.4 The Replica Symmetry Breaking and Droplet/Scaling Scenarios 194 *7.5 The Parisi Overlap Distribution 197 *7.6 Self-Averaging and Non-Self-Averaging 199 *7.7 Ruling Out the Standard RSB Scenario 201 *7.8 Chaotic Size Dependence and Metastates 203 *7.9 A New RSB Scenario 206 *7.10 Two More (Relatively) New Scenarios 211 *7.11 Why Should the SK Model Behave Differently from the EA Model? 214 *7.12 Summary: Where Do We Stand? 216 8. Are Spin Glasses Complex Systems? 218 *8.1 Three Foundational Papers 219 *8.2 Spin Glasses as a Bridge to Somewhere 227 *8.3 Modern Viewpoints on Complexity 228 *8.4 Spin Glasses: Old, New, and Quasi-Complexity 233 Notes 239 Glossary 265 Bibliography 285 Index 309

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Book SynopsisPrinceton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. This volume gathers papers from internationally renowned mathematicians, many of whom have been Stein's students.Table of ContentsPreface ix Chapter 1 Selected Theorems by Eli Stein 1 Charles Fefferman Chapter 2 Eli's Impact: A Case Study 35 Charles Fefferman Chapter 3 On Oscillatory Integral Operators in Higher Dimensions 47 Jean Bourgain Chapter 4 Holder Regularity for Generalized Master Equations with Rough Kernels 63 Luis Caffarelli and Luis Silvestre Chapter 5 Extremizers of a Radon Transform Inequality 84 Michael Christ Chapter 6 Should We Solve Plateau's Problem Again? 108 Guy David Chapter 7 Averages along Polynomial Sequences in Discrete Nilpotent Lie Groups: Singular Radon Transforms 146 Alexandru D. Ionescu, Akos Magyar, and Stephen Wainger Chapter 8 Internal DLA for Cylinders 189 David Jerison, Lionel Levine, and Scott Sheffield Chapter 9 The Energy Critical Wave Equation in 3D 215 Carlos Kenig Chapter 10 On the Bounded L2 Curvature Conjecture 224 Sergiu Klainerman Chapter 11 On Div-Curl for Higher Order 245 Loredana Lanzani and Andrew S. Raich Chapter 12 Square Functions and Maximal Operators Associated with Radial Fourier Multipliers 273 Sanghyuk Lee, Keith M. Rogers, and Andreas Seeger Chapter 13 Problems of Harmonic Analysis Related to Finite-Type Hypersurfaces in 3, and Newton Polyhedra 303 Detlef Muller Chapter 14 Multi-Linear Multipliers Associated to Simplexes of Arbitrary Length 346 Camil Muscalu, Terence Tao, and Christoph Thiele Chapter 15 Diagonal Estimates for Bergman Kernels in Monomial-Type Domains 402 Alexander Nagel and Malabika Pramanik Chapter 16 On the Singularities of the Pluricomplex Green's Function 419 D. H. Phong and Jacob Sturm Chapter 17 Smoothness of Spectral Multipliers and Convolution Kernels in Nilpotent Gelfand Pairs 436 Fulvio Ricci Chapter 18 On Eigenfunction Restriction Estimates and L4-Bounds for Compact Surfaces with Nonpositive Curvature 447 Christopher D. Sogge and Steve Zelditch List of Contributors 463 Index 465

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Book SynopsisThe author 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.Trade Review"Feldman succeeds in introducing the reader to the world of dynamic systems and the, almost mythical, chaos that they can produce."---Adhemar Bultheel, European Mathematical Society"[A] gentle and loving introduction to dynamical systems. . . . Chaos and Dynamical Systems is a book for everyone from the layman to the expert."---David S. Mazel, MAA Reviews

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Book SynopsisQuaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quateTrade ReviewOne of Choice's Outstanding Academic Titles for 2015 "Rodman fills a void in the monographic literature with this work."--Choice "The book is self-contained and well organized... Full and detailed proofs are supplied. Another exciting point is the presence of many open problems throughout the book."--Gisele C. Ducati, MatchSciNetTable of Contents*FrontMatter, pg. i*Contents, pg. vii*Preface, pg. xi*Chapter One. Introduction, pg. 1*Chapter Two. The algebra of quaternions, pg. 9*Chapter Three. Vector spaces and matrices: Basic theory, pg. 28*Chapter Four. Symmetric matrices and congruence, pg. 64*Chapter Five. Invariant subspaces and Jordan form, pg. 83*Chapter Six. Invariant neutral and semidefinite subspaces, pg. 131*Chapter Seven. Smith form and Kronecker canonical form, pg. 153*Chapter Eight. Pencils of hermitian matrices, pg. 172*Chapter Nine. Skewhermitian and mixed pencils, pg. 194*Chapter Ten. Indefinite inner products: Conjugation, pg. 228*Chapter Eleven. Matrix pencils with symmetries: Nonstandard involution, pg. 261*Chapter Twelve. Mixed matrix pencils: Nonstandard involutions, pg. 279*Chapter Thirteen. Indefinite inner products: Nonstandard involution, pg. 300*Chapter Fourteen. Matrix equations, pg. 328*Chapter Fifteen. Appendix: Real and complex canonical forms, pg. 339*Bibliography, pg. 353*Index, pg. 361

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Book SynopsisTrade Review"Nahin includes gems from all over mathematics, ranging from engineering applications to beautiful pure-mathematical identities... It would be good to have more books like this."--Timothy Gowers, Nature "Nahin's tale of the formula e[pi] i+1=0, which links five of the most important numbers in mathematics, is remarkable. With a plethora of historical and anecdotal material and a knack for linking events and facts, he gives the reader a strong sense of what drove mathematicians like Euler."--Matthew Killeya, New Scientist "It is very difficult to sum up the greatness of Euler... This excellent book goes a long way to explaining the kind of mathematician he really was."--Steve Humble, Mathematics Today "What a treasure of a book this is! This is the fourth enthusiastic, informative, and delightful book Paul Nahin has written about the beauties of various areas of mathematics... This book is a marvelous tribute to Euler's genius and those who built upon it and would make a great present for students of mathematics, physics, and engineering and their professors."--Henry Ricardo, MAA Reviews "The heart and soul of the book are the final three chapters on Fourier series, Fourier integrals, and related engineering. One can recommend them to all applied math students for their historical development and sensible content."--Robert E. O'Malley, Jr., SIAM Review "This is a book for mathematicians who enjoy historically motivated mathematical explanations on a high mathematical level."--Eberhard Knobloch, Mathematical Reviews "It is a 'popular' book, written for a general reader with some mathematical background equivalent to a first-year undergraduate course in the UK."--Robin Wilson, London Mathematical Society NewsletterTable of Contents*FrontMatter, pg. i*Contents, pg. ix*Preface to the Paperback Edition, pg. xiii*Preface, pg. xxix*Introduction, pg. 1*Chapter 1. Complex Numbers, pg. 13*Chapter 2. Vector Trips, pg. 68*Chapter 3. The Irrationality of pi2, pg. 92*Chapter 4. Fourier Series, pg. 114*Chapter 5. Fourier Integrals, pg. 188*Chapter 6. Electronics and -1, pg. 275*Euler: The Man and the Mathematical Physicist, pg. 324*Notes, pg. 347*Acknowledgments, pg. 375*Index, pg. 377

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Book SynopsisFor more than two thousand years some familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person.Trade ReviewFrom the reviews: "Many teachers and students could benefit from thinking about its unique perspectives." ---SIAM Review "Giaquinta . . . and Modica . . . present the differential and integral calculus for real functions of a single real variable from a sophisticated point of view. Their book is far better suited as a source of unique perspectives on the logical development of calculus ideas and their physical applications . . . [so that] it will better serve the experienced calculus instructor seeking new approaches to familiar material, or the student who has already mastered the basic ideas and techniques of calculus but wishes to see it all again, 'done right,' . . . Highly recommended." ---CHOICE "This self-contained book introduces the main ideas and fundamental methods of mathematical analysis without loosing sight of the context in which it was developed and the role played in science, particularly in physics…. Each chapter has a short summary where the most important facts discussed are collected. There is also a large number of exercises inserted at various points into the text…. The book is meant principally for students in mathematics, physics, engineering, and computer science, but it can be used at technological and scientific faculties by anyone who wants to approach these topics." ---ZENTRALBLATT MATH "The presentation of the theory is clearly arranged, all theorems have rigorous proofs and every chapter closes with a summing up of the results and exericeses with different requirements. The aim of the authors was very successful[ly] realized, and this book is excellently suitable for students in mathematics, physics, engineering, computer science and all students of technological and scientific faculties." ---Journal of Analysis and its Applications "This is the first of several volumes on analysis by these authors. … It is essentially a calculus book at the advanced undergraduate level, or an analysis textbook for someone who has never seen calculus before. … I like this book … ." (Warren Johnson, The Mathematical Association of America, October, 2009)Table of ContentsNumbers, Functions and their Graphs * Limits and Continuity * The Fundamental Ideas of the Differential and Integral Calculus * The Calculus of Derivatives and Integrals * Further Developments in Calculus * Towards Differential Equations and Minimum Principles * Bibliographical Notes * Index

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Book SynopsisIn particular, if f and j are x2 x 2 2 2 both O(e- / ), then f = j = Ae- / , where A is a constant;Trade Review"This nicely written book by Thangavelu is concerned with extensions of Hardy's theorem to settings that arise from noncommutative harmonic analysis.... Each chapter contains several applications to the heat equation in various settings and ends with an extensive presentation of related topics, current research, detailed references to the literature, and lists of open problems. This makes the book an invaluable resource for graduate students and researchers in harmonic analysis and applied mathematics." —SIAM Review "…Each chapter ends with useful notes and open problems. Everything is written in sufficient detail to benefit specialized interested readers…" —MATHEMATICAL REVIEWS "The authoer discusses inthe present book the original theorem of Hardy and some of its generaliztions and its connections to noncommunitave analysis, harmonic analysis and special functions. First Hardy's theorem for the Euclidian Fourier transform is treated, and a theorem of Beurling and Hömander Subsequently Hardy's theorem is dicussed for the Fourier transfom on the Heisenberg group. finally the author discusses generaliztions of Hardy's theorem involving the Helgason Fourier transform for rank one symmetric spaces and for H-type groups. This unique book will be of great value for readers interested in this branch of analysis." ---Monatshefte für MathematikTable of Contents1 Euclidean Spaces.- 1.1 Fourier transform on L1(?n).- 1.2 Hermite functions and L2 theory.- 1.3 Spherical harmonics and symmetry properties.- 1.4 Hardy’s theorem on ?n.- 1.5 Beurling’s theorem and its consequences.- 1.6 Further results and open problems.- 2 Heisenberg Groups.- 2.1 Heisenberg group and its representations.- 2.2 Fourier transform on Hn.- 2.3 Special Hermite functions.- 2.4 Fourier transform of radial functions.- 2.5 Unitary group and spherical harmonics.- 2.6 Spherical harmonics and the Weyl transform.- 2.7 Weyl correspondence of polynomials.- 2.8 Heat kernel for the sublaplacian.- 2.9 Hardy’s theorem for the Heisenberg group.- 2.10 Further results and open problems.- 3 Symmetric Spaces of Rank 1.- 3.1 A Riemannian space associated to Hn.- 3.2 The algebra of radial functions on S.- 3.3 Spherical Fourier transform.- 3.4 Helgason Fourier transform.- 3.5 Hecke-Bochner formula for the Helgason Fourier transform.- 3.6 Jacobi transforms.- 3.7 Estimating the heat kernel.- 3.8 Hardy’s theorem for the Helgason Fourier transform.- 3.9 Further results and open problems.

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Book Synopsis1. Real Numbers and Natural Numbers.- 1.1 Introduction.- 1.2 The Axiomatic Approach to Real Numbers.- 1.3 Natural Numbers.- 1.4 Summing Up.- 1.5 Exercises.- 2. Sequences of Real Numbers.- 2.1 Sequences.- 2.2 Equivalent Formulations of the Continuity Axiom.- 2.3 Limits of Sequences and Continuity.- 2.4 Some Special Sequences.- 2.5 An Alternative Definition of Exponentials and Logarithms.- 2.6 Summing Up.- 2.7 Exercises.- 3. Integer Numbers: Congruences, Counting and Infinity.- 3.1 Congruences.- 3.2 Combinatorics.- 3.3 Infinity.- 3.4 Summing Up.- 3.5 Exercises.- 4. Complex Numbers.- 4.1 Complex Numbers.- 4.2 Sequences of Complex Numbers.- 4.3 Some Elementary Applications.- 4.4 Summing Up.- 4.5 Exercises.- 5. Polynomials, Rational Functions and Trigonometric Polynomials.- 5.1 Polynomials.- 5.2 Solutions of Polynomial Equations.- 5.3 Rational Functions.- 5.4 Sinusoidal Functions and Their Sums.- 5.5 Summing Up.- 5.6 Exercises.- 6. Series.- 6.1 Basic Facts.- 6.2 Taylor Series, e and ?.- 6.3 Series of Nonnegative Terms.- 6.4 Series of Terms of Arbitrary Sign.- 6.5 Series of Products.- 6.6 Products of Series.- 6.7 Rearrangements.- 6.8 Summing Up.- 6.9 Exercises.- 7. Power Series.- 7.1 Basic Theory.- 7.2 Further Results.- 7.3 Some Applications.- 7.4 Further Applications.- 7.5 Summing Up.- 7.6 Exercises.- 8. Discrete Processes.- 8.1 Recurrences.- 8.2 One-Dimensional Dynamical Systems.- 8.3 Two-Dimensional Dynamical Systems.- 8.4 Exercises.- A. Mathematicians and Other Scientists.- B. Bibliographical Notes.- C. Index.Trade Review"This self-contained book aims to introduce the main ideas for studying approximation processes, more generally discrete processes at graduate level. The use of computers induces a growing need for studying discrete processes.... A key feature this lively yet rigorous and systematic treatment is the historical accounts of ideas and methods of the subject. Ideas in mathematics develop in cultural, historical and economical contexts, thus the authors made brief accounts of those aspects and used a large number of beautiful illustrations.... Each chapter has a short summary where the most important facts discussed are collected and described. There is also a large number of exercises inserted at various points into the text....The book is meant principally for graduate students in mathematics, physics, engineering, and computer science, but it can be used at technological and scientific faculties by anyone who wants to approach these topics. It may also be used in graduate seminars and courses or as a reference text by mathematicians, physicists, and engineers." —Zentralblatt MATH"Mathematical Analysis does contain a substantial amount of material that is unusual in terms of an introductory text in real analysis…These are all interesting topics that have gained increasing importance in modern applications of mathematics, albeit outside the traditional area of analysis. It is very nice to have these topics developed outside a specialized textbook, in, e.g., combinatorics, dynamical systems, or number theory. The authors do a very good job presenting this material…Mathematical Analysis includes substantial amounts of historical background…The book also contains a lot of examples…I can happily recommend Mathematical Analysis as a good resource for instructors of introductory analysis courses, especially in terms of providing some unusual applications of analysis and developments of some basic classic topics that are often shortchanged in standard texts." —SIAM Review“This is the second volume of a series on analysis. … a real hodgepodge that could only be the primary textbook for a course specifically based on it. … In this series Giaquinta and Modica have set themselves the formidable task of constructing from scratch an analysis sequence of several years length. … they have more regard for classical topics and arguments than most authors writing analysis books today. … I enjoyed reading this volume … .” (Warren Johnson, The Mathematical Association of America, January, 2010) Table of ContentsPreface * Real Numbers and Natural Numbers * Sequences of Real Numbers * Integer Numbers: Congruences, Counting and Infinity * Complex Numbers * Polynomials, Rational Functions and Trigonometric Polynomials * Series * Power Series * Discrete Processes * Mathematicans and Other Scientists * Bibliographical Notes * Index

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