Calculus and mathematical analysis Books

Book SynopsisTrade ReviewOne of the Top 10 Technical Books on Financial Engineering by Financial Engineering News for 2006 Praise for the previous edition: "This book provides a state-of-the-art discussion of the three main categories of risk in financial markets, market risk, ... credit risk ... and operational risk... This is a high level, but well-written treatment, rigorous (sometimes succinct), complete with theorems and proofs."--D.L. McLeish, Short Book Reviews of the International Statistical Institute Praise for the previous edition: "A great summary of the latest techniques available within quantitative risk measurement... [I]t is an excellent text to have on the shelf as a reference when your day job covers the whole spectrum of quantitative techniques in risk management."--Financial Engineering News Praise for the previous edition: "Alexander McNeil, Rudiger Frey and Paul Embrechts have written a beautiful book... [T]here is no book that can provide the type of rigorous, detailed, well balanced and relevant coverage of quantitative risk management topics that Quantitative Risk Management: Concepts, Techniques, and Tools offers... I believe that this work may become the book on quantitative risk management... [N]o book that I know of can provide better guidance."--Dr. Riccardo Rebonato, Global Association of Risk Professionals (GARP) Review Praise for the previous edition: "This is a very impressive book on a rapidly growing field. It certainly helps to discover the forest in an area where a lot of trees are popping up daily."--Hans Buhlmann, SIAM Review Praise for the previous edition: "This book is a compendium of the statistical arrows that should be in any quantitative risk manager's quiver. It includes extensive discussion of dynamic volatility models, extreme value theory, copulas and credit risk. Academics, PhD students and quantitative practitioners will find many new and useful results in this important volume."--Robert F. Engle III, 2003 Nobel Laureate in Economic Sciences, Michael Armellino Professor in the Management of Financial Services at New York University's Stern School of Business Praise for the previous edition: "Quantitative Risk Management can be highly recommended to anyone looking for an excellent survey of the most important techniques and tools used in this rapidly growing field."--Holger Drees, Risk Praise for the previous edition: "Quantitative Risk Management is highly recommended for financial regulators. The statistical and mathematical tools facilitate a better understanding of the strengths and weaknesses of a useful range of advanced risk-management concepts and models, while the focus on aggregate risk enhances the publication's value to banking and insurance supervisors."--Hans Blommestein, Financial Regulator Praise for the previous edition: "This book provides a framework and a useful toolkit for analysis of a wide variety of risk management problems. Common pitfalls are pointed out, and mathematical sophistication is used in pursuit of useful and usable solutions. Every financial institution has a risk management department that looks at aggregated portfolio-wide risks on longer time scales, and at risk exposure to large, or extreme, market movements. Risk managers are always on the lookout for good techniques to help them do their jobs. This very good book provides these techniques and addresses an important, and under-developed, area of practical research."--Martin Baxter, Nomura International

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Book SynopsisCalculus Made Easy by Silvanus P. Thompson and Martin Gardner has long been the most popular calculus primer. This major revision of the classic math text makes the subject at hand still more comprehensible to readers of all levels. With a new introduction, three new chapters, modernized language and methods throughout, and an appendix of challenging and enjoyable practice problems, Calculus Made Easy has been thoroughly updated for the modern reader.

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Book SynopsisTable of ContentsChapter 1: The Real and Complex Number SystemsIntroductionOrdered SetsFieldsThe Real FieldThe Extended Real Number SystemThe Complex FieldEuclidean SpacesAppendixExercisesChapter 2: Basic TopologyFinite, Countable, and Uncountable SetsMetric SpacesCompact SetsPerfect SetsConnected SetsExercisesChapter 3: Numerical Sequences and SeriesConvergent SequencesSubsequencesCauchy SequencesUpper and Lower LimitsSome Special SequencesSeriesSeries of Nonnegative TermsThe Number eThe Root and Ratio TestsPower SeriesSummation by PartsAbsolute ConvergenceAddition and Multiplication of SeriesRearrangementsExercisesChapter 4: ContinuityLimits of FunctionsContinuous FunctionsContinuity and CompactnessContinuity and ConnectednessDiscontinuitiesMonotonic FunctionsInfinite Limits and Limits at InfinityExercisesChapter 5: DifferentiationThe Derivative of a Real FunctionMean Value TheoremsThe Continuity of DerivativesL'Hospital's RuleDerivatives of Higher-OrderTaylor's TheoremDifferentiation of Vector-valued FunctionsExercisesChapter 6: The Riemann-Stieltjes IntegralDefinition and Existence of the IntegralProperties of the IntegralIntegration and DifferentiationIntegration of Vector-valued FunctionsRectifiable CurvesExercisesChapter 7: Sequences and Series of FunctionsDiscussion of Main ProblemUniform ConvergenceUniform Convergence and ContinuityUniform Convergence and IntegrationUniform Convergence and DifferentiationEquicontinuous Families of FunctionsThe Stone-Weierstrass TheoremExercisesChapter 8: Some Special FunctionsPower SeriesThe Exponential and Logarithmic FunctionsThe Trigonometric FunctionsThe Algebraic Completeness of the Complex FieldFourier SeriesThe Gamma FunctionExercisesChapter 9: Functions of Several VariablesLinear TransformationsDifferentiationThe Contraction PrincipleThe Inverse Function TheoremThe Implicit Function TheoremThe Rank TheoremDeterminantsDerivatives of Higher OrderDifferentiation of IntegralsExercisesChapter 10: Integration of Differential FormsIntegrationPrimitive MappingsPartitions of UnityChange of VariablesDifferential FormsSimplexes and ChainsStokes' TheoremClosed Forms and Exact FormsVector AnalysisExercisesChapter 11: The Lebesgue TheorySet FunctionsConstruction of the Lebesgue MeasureMeasure SpacesMeasurable FunctionsSimple FunctionsIntegrationComparison with the Riemann IntegralIntegration of Complex FunctionsFunctions of Class L2ExercisesBibliographyList of Special SymbolsIndex

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Book SynopsisTable of ContentsPart I: MECHANICS. 1. Physics and Measurement. 2. Motion in One Dimension. 3. Vectors. 4. Motion in Two Dimensions. 5. The Laws of Motion. 6. Circular Motion and Other Applications of Newton's Laws. 7. Energy of a System. 8. Conservation of Energy. 9. Linear Momentum and Collisions. 10. Rotation of a Rigid Object About a Fixed Axis. 11. Angular Momentum. 12. Static Equilibrium and Elasticity. 13. Universal Gravitation. 14. Fluid Mechanics. Part II: OSCILLATIONS AND MECHANICAL WAVES. 15. Oscillatory Motion. 16. Wave Motion. 17. Superposition and Standing Waves. Part III: THERMODYNAMICS. 18. Temperature. 19. Heat and the First Law of Thermodynamics. 20. The Kinetic Theory of Gases. 21. Heat Engines, Entropy, and the Second Law of Thermodynamics. Part IV: ELECTRICITY AND MAGNETISM. 22. Electric Fields. 23. Continuous Charge Distributions and Gauss's Law. 24. Electric Potential. 25. Capacitance and Dielectrics. 26. Current and Resistance. 27. Direct Current Circuits. 28. Magnetic Fields. 29. Sources of the Magnetic Field. 30. Faraday's Law. 31. Inductance. 32. Alternating Current Circuits. 33. Electromagnetic Waves. Part V: LIGHT AND OPTICS. 34. The Nature of Light and the Laws of Geometric Optics. 35. Image Formation. 36. Interference of Light Waves. 37. Diffraction Patterns and Polarization. Part VI: MODERN PHYSICS. 38. Relativity. 39. Introduction to Quantum Physics. 40. Quantum Mechanics. 41. Atomic Physics. 42. Molecules and Solids. 43. Nuclear Physics. 44. Particle Physics and Cosmology. APPENDICES. A. Tables. B. Mathematics Review. C. Periodic Table of the Elements. D. SI Units. Answers to Quick Quizzes and Odd-Numbered Problems. Index.

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Book SynopsisCalculus is important for first-year undergraduate students pursuing mathematics, physics, economics, engineering, and other disciplines where mathematics plays a significant role. The book provides a thorough reintroduction to calculus with an emphasis on logical development arising out of geometric intuition. The author has restructured the subject matter in the book by using Tarski''s version of the completeness axiom, introducing integration before differentiation and limits, and emphasizing benefits of monotonicity before continuity. The standard transcendental functions are developed early in a rigorous manner and the monotonicity theorem is proved before the mean value theorem. Each concept is supported by diverse exercises which will help the reader to understand applications and take them nearer to real and complex analysis.Table of ContentsIntroduction; 1. Real Numbers and Functions; 2. Integration; 3. Limits and Continuity; 4. Differentiation; 5. Techniques of Integration; 6. Mean Value Theorems and Applications; 7. Sequences and Series; 8. Taylor and Fourier Series; A. Solutions to Odd-Numbered Exercises; Bibliography; Index.

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Book SynopsisJoel Hass received his PhD from the University of California - Berkeley. He is currently a professor of mathematics at the University of California - Davis. He has coauthored widely used calculus texts as well as calculus study guides. He is currently on the editorial board of several publications, including the Notices of the American Mathematical Society. He has been a member of the Institute for Advanced Study at Princeton University and of the Mathematical Sciences Research Institute, and he was a Sloan Research Fellow. Hass's current areas of research include the geometry of proteins, three dimensional manifolds, applied math, and computational complexity. In his free time, Hass enjoys kayaking. Christopher Heil received his PhD from the University of Maryland. He is currently a professor of mathematics at the Georgia Institute of Technology. He is the author of a graduate text on analysis and a number of highly cited research survey artTable of Contents1. Functions 1.1 Functions and Their Graphs 1.2 Combining Functions; Shifting and Scaling Graphs 1.3 Trigonometric Functions 1.4 Graphing with Software Questions to Guide Your Review Practice Exercises Additional and Advanced Exercises Technology Application Projects 2. Limits and Continuity 2.1 Rates of Change and Tangent Lines to Curves 2.2 Limit of a Function and Limit Laws 2.3 The Precise Definition of a Limit 2.4 One-Sided Limits 2.5 Limits Involving Infinity; Asymptotes of Graphs 2.6 Continuity Questions to Guide Your Review Practice Exercises Additional and Advanced Exercises Technology Application Projects 3. Derivatives 3.1 Tangent Lines and the Derivative at a Point 3.2 The Derivative as a Function 3.3 Differentiation Rules 3.4 The Derivative as a Rate of Change 3.5 Derivatives of Trigonometric Functions 3.6 The Chain Rule 3.7 Implicit Differentiation 3.8 Derivatives of Inverse Functions and Logarithms 3.9 Related Rates 3.10 Linearization and Differentials Questions to Guide Your Review Practice Exercises Additional and Advanced Exercises Technology Application Projects 4. Applications of Derivatives 4.1 Extreme Values of Functions on Closed Intervals 4.2 The Mean Value Theorem 4.3 Monotonic Functions and the First Derivative Test 4.4 Concavity and Curve Sketching 4.5 Applied Optimization 4.6 Newton's Method 4.7 Antiderivatives Questions to Guide Your Review Practice Exercises Additional and Advanced Exercises Technology Application Projects 5. Integrals 5.1 Area and Estimating with Finite Sums 5.2 Sigma Notation and Limits of Finite Sums 5.3 The Definite Integral 5.4 The Fundamental Theorem of Calculus 5.5 Indefinite Integrals and the Substitution Method 5.6 Definite Integral Substitutions and the Area Between Curves Questions to Guide Your Review Practice Exercises Additional and Advanced Exercises Technology Application Projects 6. Applications of Definite Integrals 6.1 Volumes Using Cross-Sections 6.2 Volumes Using Cylindrical Shells 6.3 Arc Length 6.4 Areas of Surfaces of Revolution 6.5 Work and Fluid Forces 6.6 Moments and Centers of Mass Questions to Guide Your Review Practice Exercises Additional and Advanced Exercises Technology Application Projects 7. Transcendental Functions 7.1 Inverse Functions and Their Derivatives 7.2 Natural Logarithms 7.3 Exponential Functions 7.4 Exponential Change and Separable Differential Equations 7.5 Indeterminate Forms and L'Hôpital's Rule 7.6 Inverse Trigonometric Functions 7.7 Hyperbolic Functions 7.8 Relative Rates of Growth Questions to Guide Your Review Practice Exercises Additional and Advanced Exercises 8. Techniques of Integration 8.1 Using Basic Integration Formulas 8.2 Integration by Parts 8.3 Trigonometric Integrals 8.4 Trigonometric Substitutions 8.5 Integration of Rational Functions by Partial Fractions 8.6 Integral Tables and Computer Algebra Systems 8.7 Numerical Integration 8.8 Improper Integrals Questions to Guide Your Review Practice Exercises Additional and Advanced Exercises Technology Application Projects 9. Infinite Sequences and Series 9.1 Sequences 9.2 Infinite Series 9.3 The Integral Test 9.4 Comparison Tests 9.5 Absolute Convergence; The Ratio and Root Tests 9.6 Alternating Series and Conditional Convergence 9.7 Power Series 9.8 Taylor and Maclaurin Series 9.9 Convergence of Taylor Series 9.10 Applications of Taylor Series Questions to Guide Your Review Practice Exercises Additional and Advanced Exercises Technology Application Projects 10. Parametric Equations and Polar Coordinates 10.1 Parametrizations of Plane Curves 10.2 Calculus with Parametric Curves 10.3 Polar Coordinates 10.4 Graphing Polar Coordinate Equations 10.5 Areas and Lengths in Polar Coordinates 10.6 Conic Sections 10.7 Conics in Polar Coordinates Questions to Guide Your Review Practice Exercises Additional and Advanced Exercises Technology Application Projects 11. Vectors and the Geometry of Space 11.1 Three-Dimensional Coordinate Systems 11.2 Vectors 11.3 The Dot Product 11.4 The Cross Product 11.5 Lines and Planes in Space 11.6 Cylinders and Quadric Surfaces Questions to Guide Your Review Practice Exercises Additional and Advanced Exercises Technology Application Projects 12. Vector-Valued Functions and Motion in Space 12.1 Curves in Space and Their Tangents 12.2 Integrals of Vector Functions; Projectile Motion 12.3 Arc Length in Space 12.4 Curvature and Normal Vectors of a Curve 12.5 Tangential and Normal Components of Acceleration 12.6 Velocity and Acceleration in Polar Coordinates Questions to Guide Your Review Practice Exercises Additional and Advanced Exercises Technology Application Projects 13. Partial Derivatives 13.1 Functions of Several Variables 13.2 Limits and Continuity in Higher Dimensions 13.3 Partial Derivatives 13.4 The Chain Rule 13.5 Directional Derivatives and Gradient Vectors 13.6 Tangent Planes and Differentials 13.7 Extreme Values and Saddle Points 13.8 Lagrange Multipliers 13.9 Taylor’s Formula for Two Variables 13.10 Partial Derivatives with Constrained Variables Questions to Guide Your Review Practice Exercises Additional and Advanced Exercises Technology Application Projects 14. Multiple Integrals 14.1 Double and Iterated Integrals over Rectangles 14.2 Double Integrals over General Regions 14.3 Area by Double Integration 14.4 Double Integrals in Polar Form 14.5 Triple Integrals in Rectangular Coordinates 14.6 Applications 14.7 Triple Integrals in Cylindrical and Spherical Coordinates 14.8 Substitutions in Multiple Integrals Questions to Guide Your Review Practice Exercises Additional and Advanced Exercises Technology Application Projects 15. Integrals and Vector Fields 15.1 Line Integrals of Scalar Functions 15.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 15.3 Path Independence, Conservative Fields, and Potential Functions 15.4 Green’s Theorem in the Plane 15.5 Surfaces and Area 15.6 Surface Integrals 15.7 Stokes’ Theorem 15.8 The Divergence Theorem and a Unified Theory Questions to Guide Your Review Practice Exercises Additional and Advanced Exercises Technology Application Projects 16. First-Order Differential Equations 16.1 Solutions, Slope Fields, and Euler’s Method 16.1 Solutions, Slope Fields, and Euler’s Method 16.2 First-Order Linear Equations 16.3 Applications 16.4 Graphical Solutions of Autonomous Equations 16.5 Systems of Equations and Phase Planes Questions to Guide Your Review Practice Exercises Additional and Advanced Exercises Technology Application Projects 17. Second-Order Differential Equations (online) 17.1 Second-Order Linear Equations 17.2 Nonhomogeneous Linear Equations 17.3 Applications 17.4 Euler Equations 17.5 Power-Series Solutions 18. Complex Functions (online) 18.1 Complex Numbers 18.2 Functions of a Complex Variable 18.3 Derivatives 18.4 The Cauchy-Riemann Equations 18.5 Complex Power Series 18.6 Some Complex Functions 18.7 Conformal Maps Questions to Guide Your Review Additional and Advanced Exercises 19. Fourier Series and Wavelets (online) 19.1 Periodic Functions 19.2 Summing Sines and Cosines 19.3 Vectors and Approximation in Three and More Dimensions 19.4 Approximation of Functions 19.5 Advanced Topic: The Haar System and Wavelets Questions to Guide Your Review Additional and Advanced Exercises Appendix A A.1 Real Numbers and the Real Line A.2 Mathematical Induction A.3 Lines, Circles, and Parabolas A.4 Proofs of Limit Theorems A.5 Commonly Occurring Limits A.6 Theory of the Real Numbers A.7 Probability A.8 The Distributive Law for Vector Cross Products A.9 The Mixed Derivative Theorem and the Increment Theorem Appendix B (online) B.1 Determinants B.2 Extreme Values and Saddle Points for Functions of More than Two Variables B.3 The Method of Gradient Descent Answers to Odd-Numbered Exercises Applications Index Subject Index Credits A Brief Table of Integrals

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Book SynopsisThis is part two of a two-volume introduction to real analysis and is intended for honours undergraduates who have already been exposed to calculus. The emphasis is on rigour and on foundations. The material starts at the very beginning--the construction of the number systems and set theory--then goes on to the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and finally to the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. There are also appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of twenty-five to thirty lectures each.The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.The fourth edition incorporates a large number of additional corrections reported since the release of the third edition, as well as some additional new exercises.

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Book SynopsisCalculus is the key to much of modern science and engineering. It is the mathematical method for the analysis of things that change, and since in the natural world we are surrounded by change, the development of calculus was a huge breakthrough in the history of mathematics. But it is also something of a mathematical adventure, largely because of the way infinity enters at virtually every twist and turn...In The Calculus Story David Acheson presents a wide-ranging picture of calculus and its applications, from ancient Greece right up to the present day. Drawing on their original writings, he introduces the people who helped to build our understanding of calculus. With a step by step treatment, he demonstrates how to start doing calculus, from the very beginning.Trade ReviewA masterpiece... Packed with insights, both historical and mathematical. * Steven Strogatz, professor of mathematics, Cornell University, and author of The Joy of X and Infinite Powers *This is the book on calculus I wish I'd written. It's a beautifully simple, friendly guide that's bursting at the seams with glorious, persuasive explanations as to why calculus is one of the most powerful ideas ever conceived by mankind. * Hannah Fry, Broadcaster, lecturer, and author of The Mathematics of Love *A splendid little book ... accessible to a very wide audience ... The book is highly recommended. * Adam McBride, Mathematical Gazette *A remarkably expansive and frictionless tour of mathematical history and theory... The calculus story is no textbook... It is the antithesis of the dreary way calculus is too often taught at schools and universities... a supplement for a high school student, the parents of such a student, or an adult wishing to reacquaint herself painlessly with material long forgotten. * Henrik Latter, Plus *This is a very readable book... It offers an illuminating perspective on calculus... A very enjoyable book for the layperson or the user of calculus. * Alex Chaplin, School Science Review *Wish I'd had it as a maths student! * Tim Harford, Undercover Economist *Another wonderful book. * Mark McCartney, LMS Newsletter *A very clear explanation of calculus ([I] wish I'd had it as a maths student!) along with some history of the subject. * Tim Harford, The Undercover Economist *Superb introduction to calculus that should be in every young mathematician's bookcase. * Peter Ransom, Symmetry Plus *Don't panic if your mathematical muscles appear to have withered away (or you never truly cracked differentiation), David Acheson's The Calculus Story could be just the thing... A roller-coaster read, constantly climbing and diving through the wonderful world of calculus... There's something for everyone, from the inexperienced integrator to the seasoned solver of equations... His enthusiasm for calculus is almost palpable. * Timothy Revell, New Scientist *Dazzling. * Matthew Reisz, Times Higher Education *I would have killed for this book when I was 13 ... he [David Acheson] belongs in the league of great authors of popular works on mathematics. * George Matthews, Mathematics Today *A worthy successor to 1089 and All That. * Adhemar Bult heel, European Mathematical Society *A simple guide to calculus - where it came from, how it works, what it's good for, and where it went. Brief, informative, charming, and a model of clarity. Ideal motivation for beginners, and recommended to anyone who wonders what the subject is about. * Ian Stewart, author of Seventeen Equations that Changed the World *This wide-ranging picture of calculus and its applications, from antiquity to the present, reveals the method as both the key to much of modern science and engineering, and something of a mathematical adventure. * Science *Acheson offers a much-needed short account of the big picture of calculus as a whole, illustrated with examples and reproductions from historic publications [...] Short pages, many illustrations, and a sense of telling a big story contribute to the success of the book. * Paul J. Campbell, Mathematical Magazine *Table of ContentsREFERENCES; FURTHER READING; INDEX

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Book SynopsisTable of Contents 1. Functions 2. Limits 3. Derivatives 4. Applications of the Derivative 5. Integration 6. Applications of Integration 7. Integration Techniques 8. Sequences and Infinite Series 9. Power Series 10. Parametric and Polar Curves 11. Vectors and Vector-Valued Functions 12. Functions of Several Variables 13. Multiple Integration 14. Vector Calculus Appendix A. Algebra Review Appendix B. Proofs of Selected Theorems D1. Differential Equations (online) D1.1 Basic Ideas D1.2 Direction Fields and Euler’s Method D1.3 Separable Differential Equations D1.4 Special First-Order Differential Equations D1.5 Modeling with Differential Equations D2. Second-Order Differential Equations (online) D2.1 Basic Ideas D2.2 Linear Homogeneous Equations D2.3 Linear Nonhomogeneous Equations D2.4 Applications D2.5 Complex Forcing Functions

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Book SynopsisPerfected over three editions and more than forty years, this field- and classroom-tested reference: * Uses the method of maximum likelihood to a large extent to ensure reasonable, and in some cases optimal procedures. * Treats all the basic and important topics in multivariate statistics. * Adds two new chapters, along with a number of new sections. * Provides the most methodical, up-to-date information on MV statistics available.Trade Review"…suitable for a graduate-level course on multivariate analysis…an important reference on the bookshelves of many scientific researchers and most practicing statisticians." (Journal of the American Statistical Association, September 2004) “…really well written. The edition will be certainly welcomed…” (Zentralblatt Math, Vo.1039, No.08, 2004) "…a wonderful textbook…that covers the mathematical theory of multivariate statistical analysis…" (Clinical Chemistry, Vol. 50, No. 2, May 2004) "...remains an authoritative work that can still be highly recommended..." (Short Book Reviews, 2004) "...still a very serious and comprehensive book on the statistical theory of multivariate analysis." (Technometrics, Vol. 46, No. 1, February 2004) “...remains a mathematically rigorous development of statistical methods for observations consisting of several measurements or characteristics of each subject and a study of their properties.” (Quarterly of Applied Mathematics, Vol. LXI, No. 4, December 2003)Table of ContentsPreface to the Third Edition. Preface to the Second Edition. Preface to the First Edition. 1. Introduction. 2. The Multivariate Normal Distribution. 3. Estimation of the Mean Vector and the Covariance Matrix. 4. The Distributions and Uses of Sample Correlation Coefficients. 5. The Generalized T2-Statistic. 6. Classification of Observations. 7. The Distribution of the Sample Covariance Matrix and the Sample Generalized Variance. 8. Testing the General Linear Hypothesis: Multivariate Analysis of Variance 9. Testing Independence of Sets of Variates. 10. Testing Hypotheses of Equality of Covariance Matrices and Equality of Mean Vectors and Covariance Matrices. 11. Principal Components. 12. Cononical Correlations and Cononical Variables. 13. The Distributions of Characteristic Roots and Vectors. 14. Factor Analysis. 15. Pattern of Dependence; Graphical Models. Appendix A: Matrix Theory. Appendix B: Tables. References. Index.

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Book SynopsisThis well-respected book introduces readers to the theory and application of modern numerical approximation techniques. Providing an accessible treatment that only requires a calculus prerequisite, the authors explain how, why, and when approximation techniques can be expected to work-and why, in some situations, they fail. A wealth of examples and exercises develop readers' intuition, and demonstrate the subject's practical applications to important everyday problems in math, computing, engineering, and physical science disciplines. Three decades after it was first published, Burden, Faires, and Burden's NUMERICAL ANALYSIS remains the definitive introduction to a vital and practical subject.Table of Contents1. MATHEMATICAL PRELIMINARIES AND ERROR ANALYSIS. Review of Calculus. Round-off Errors and Computer Arithmetic. Algorithms and Convergence. Numerical Software and Chapter Summary. 2. SOLUTIONS OF EQUATIONS IN ONE VARIABLE. The Bisection Method. Fixed-Point Iteration. Newton's Method and Its Extensions. Error Analysis for Iterative Methods. Accelerating Convergence. Zeros of Polynomials and M��ller's Method. Numerical Software and Chapter Summary. 3. INTERPOLATION AND POLYNOMIAL APPROXIMATION. Interpolation and the Lagrange Polynomial. Data Approximation and Neville's Method. Divided Differences. Hermite Interpolation. Cubic Spline Interpolation. Parametric Curves. Numerical Software and Chapter Summary. 4. NUMERICAL DIFFERENTIATION AND INTEGRATION. Numerical Differentiation. Richardson's Extrapolation. Elements of Numerical Integration. Composite Numerical Integration. Romberg Integration. Adaptive Quadrature Methods. Gaussian Quadrature. Multiple Integrals. Improper Integrals. Numerical Software and Chapter Summary. 5. INITIAL-VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS. The Elementary Theory of Initial-Value Problems. Euler's Method. Higher-Order Taylor Methods. Runge-Kutta Methods. Error Control and the Runge-Kutta-Fehlberg Method. Multistep Methods. Variable Step-Size Multistep Methods. Extrapolation Methods. Higher-Order Equations and Systems of Differential Equations. Stability. Stiff Differential Equations. Numerical Software and Chapter Summary. 6. DIRECT METHODS FOR SOLVING LINEAR SYSTEMS. Linear Systems of Equations. Pivoting Strategies. Linear Algebra and Matrix Inversion. The Determinant of a Matrix. Matrix Factorization. Special Types of Matrices. Numerical Software and Chapter Summary. 7. ITERATIVE TECHNIQUES IN MATRIX ALGEBRA. Norms of Vectors and Matrices. Eigenvalues and Eigenvectors. The Jacobi and Gauss-Siedel Iterative Techniques. Relaxation Techniques for Solving Linear Systems. Error Bounds and Iterative Refinement. The Conjugate Gradient Method. Numerical Software and Chapter Summary. 8. APPROXIMATION THEORY. Discrete Least Squares Approximation. Orthogonal Polynomials and Least Squares Approximation. Chebyshev Polynomials and Economization of Power Series. Rational Function Approximation. Trigonometric Polynomial Approximation. Fast Fourier Transforms. Numerical Software and Chapter Summary. 9. APPROXIMATING EIGENVALUES. Linear Algebra and Eigenvalues. Orthogonal Matrices and Similarity Transformations. The Power Method. Householder's Method. The QR Algorithm. Singular Value Decomposition. Numerical Software and Chapter Summary. 10. NUMERICAL SOLUTIONS OF NONLINEAR SYSTEMS OF EQUATIONS. Fixed Points for Functions of Several Variables. Newton's Method. Quasi-Newton Methods. Steepest Descent Techniques. Homotopy and Continuation Methods. Numerical Software and Chapter Summary. 11. BOUNDARY-VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS. The Linear Shooting Method. The Shooting Method for Nonlinear Problems. Finite-Difference Methods for Linear Problems. Finite-Difference Methods for Nonlinear Problems. The Rayleigh-Ritz Method. Numerical Software and Chapter Summary. 12. NUMERICAL SOLUTIONS TO PARTIAL DIFFERENTIAL EQUATIONS. Elliptic Partial Differential Equations. Parabolic Partial Differential Equations. Hyperbolic Partial Differential Equations. An Introduction to the Finite-Element Method. Numerical Software and Chapter Summary. Bibliography. Answers to Selected Exercises.

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Book SynopsisBeginning with the basic facts of functional analysis, this title looks at Banach spaces, Lp spaces, and distribution theory, and highlights their roles in harmonic analysis. It uses the Baire category theorem to illustrate several points, including the existence of Besicovitch sets.Trade Review"Functional Analysis by Elias Stein and Rami Shakarchi is a fast-paced book on functional analysis and related topics. By page 60, you've had a decent course in functional analysis and you've got 360 pages left."--John D. Cook, Endeavour blog "Characteristically, Stein and Shakarchi reward readers for hard work by making the material pay off."--Choice "This excellent book ends with a proof of the continuity of the averaging operator and applications to the determination of remainder terms in asymptotic formulas for the counting function of lattice points. Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics."--Stevan Pilipovic, MathSciNet, Mathematical Reviews on the Web "This book is accessible for graduate students. Moreover, it plays the role of an instructional book in various branches of mathematical analysis, geometry, probability, and partial differential equations. In most mathematical centers one cannot expect that such lectures will be offered as a semester-long course to students, but both students and teachers have here an excellent guide for learning and teaching the topics presented in this volume... Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics."--Stevan Pilipovit, Mathematical Reviews

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Book SynopsisPresents a novel approach to conducting meta-analysis using structural equation modeling. Structural equation modeling (SEM) and meta-analysis are two powerful statistical methods in the educational, social, behavioral, and medical sciences. They are often treated as two unrelated topics in the literature. This book presents a unified framework on analyzing meta-analytic data within the SEM framework, and illustrates how to conduct meta-analysis using the metaSEM package in the R statistical environment. Meta-Analysis: A Structural Equation Modeling Approach begins by introducing the importance of SEM and meta-analysis in answering research questions. Key ideas in meta-analysis and SEM are briefly reviewed, and various meta-analytic models are then introduced and linked to the SEM framework. Fixed-, random-, and mixed-effects models in univariate and multivariate meta-analyses, three-level meta-analysis, and meta-analytic structural equation modeling, areTrade Review"This book will be a valuable resource for statistical and academic researchers and graduate students carrying out meta-analyses, and will also be useful to researchers and statisticians using SEM in biostatistics. cover, would sit well on the bookshelves of those interested in this increasingly important field of scientific endeavour." (Zentralblatt MATH, 1 June 2015)Table of ContentsPreface xiii Acknowledgments xv List of abbreviations xvii List of figures xix List of tables xxi 1 Introduction 1 1.1 What is meta-analysis? 1 1.2 What is structural equation modeling? 2 1.3 Reasons for writing a book on meta-analysis and structural equation modeling 3 1.4 Outline of the following chapters 6 1.5 Concluding remarks and further readings 8 2 Brief review of structural equation modeling 13 2.1 Introduction 13 2.2 Model specification 14 2.3 Common structural equation models 18 2.4 Estimation methods, test statistics, and goodness-of-fit indices 25 2.5 Extensions on structural equation modeling 38 2.6 Concluding remarks and further readings 42 3 Computing effect sizes for meta-analysis 48 3.1 Introduction 48 3.2 Effect sizes for univariate meta-analysis 50 3.3 Effect sizes for multivariate meta-analysis 57 3.4 General approach to estimating the sampling variances and covariances 60 3.5 Illustrations Using R 68 3.6 Concluding remarks and further readings 78 4 Univariate meta-analysis 81 4.1 Introduction 81 4.2 Fixed-effects model 83 4.3 Random-effects model 87 4.4 Comparisons between the fixed- and the random-effects models 93 4.5 Mixed-effects model 96 4.6 Structural equation modeling approach 100 4.7 Illustrations using R 105 4.8 Concluding remarks and further readings 116 5 Multivariate meta-analysis 121 5.1 Introduction 121 5.2 Fixed-effects model 124 5.3 Random-effects model 127 5.4 Mixed-effects model 134 5.5 Structural equation modeling approach 136 5.6 Extensions: mediation and moderation models on the effect sizes 140 5.7 Illustrations using R 145 5.8 Concluding remarks and further readings 174 6 Three-level meta-analysis 179 6.1 Introduction 179 6.2 Three-level model 183 6.3 Structural equation modeling approach 188 6.4 Relationship between the multivariate and the three-level meta-analyses 195 6.5 Illustrations using R 200 6.6 Concluding remarks and further readings 210 7 Meta-analytic structural equation modeling 214 7.1 Introduction 214 7.2 Conventional approaches 218 7.3 Two-stage structural equation modeling: fixed-effects models 223 7.4 Two-stage structural equation modeling: random-effects models 233 7.5 Related issues 235 7.6 Illustrations using R 244 7.7 Concluding remarks and further readings 273 8 Advanced topics in SEM-based meta-analysis 279 8.1 Restricted (or residual) maximum likelihood estimation 279 8.2 Missing values in the moderators 289 8.3 Illustrations using R 294 8.4 Concluding remarks and further readings 309 9 Conducting meta-analysis with Mplus 313 9.1 Introduction 313 9.2 Univariate meta-analysis 314 9.3 Multivariate meta-analysis 327 9.4 Three-level meta-analysis 346 9.5 Concluding remarks and further readings 353 A A brief introduction to R, OpenMx, and metaSEM packages 356 A.1 R 357 A.2 OpenMx 362 A.3 metaSEM 364 References 368 Index 369

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Book SynopsisAn accessible guide to the multivariate time series tools used in numerous real-world applications Multivariate Time Series Analysis: With R and Financial Applications is the much anticipated sequel coming from one of the most influential and prominent experts on the topic of time series.Table of ContentsPreface xv Acknowledgements xvii 1 Multivariate Linear Time Series 1 1.1 Introduction, 1 1.2 Some Basic Concepts, 5 1.3 Cross-Covariance and Correlation Matrices, 8 1.4 Sample CCM, 9 1.5 Testing Zero Cross-Correlations, 12 1.6 Forecasting, 16 1.7 Model Representations, 18 1.8 Outline of the Book, 22 1.9 Software, 23 Exercises, 23 2 Stationary Vector Autoregressive Time Series 27 2.1 Introduction, 27 2.2 VAR(1) Models, 28 2.3 VAR(2) Models, 37 2.4 VAR(p) Models, 41 2.5 Estimation, 44 2.6 Order Selection, 61 2.7 Model Checking, 66 2.8 Linear Constraints, 80 2.9 Forecasting, 82 2.10 Impulse Response Functions, 89 2.11 Forecast Error Variance Decomposition, 96 2.12 Proofs, 98 Exercises, 100 3 Vector Autoregressive Moving-Average Time Series 105 3.1 Vector MA Models, 106 3.2 Specifying VMA Order, 112 3.3 Estimation of VMA Models, 113 3.4 Forecasting of VMA Models, 126 3.5 VARMA Models, 127 3.6 Implications of VARMA Models, 139 3.7 Linear Transforms of VARMA Processes, 141 3.8 Temporal Aggregation of VARMA Processes, 144 3.9 Likelihood Function of a VARMA Model, 146 3.10 Innovations Approach to Exact Likelihood Function, 155 3.11 Asymptotic Distribution of Maximum Likelihood Estimates, 160 3.12 Model Checking of Fitted VARMA Models, 163 3.13 Forecasting of VARMA Models, 164 3.14 Tentative Order Identification, 166 3.15 Empirical Analysis of VARMA Models, 176 3.16 Appendix, 192 Exercises, 194 4 Structural Specification of VARMA Models 199 4.1 The Kronecker Index Approach, 200 4.2 The Scalar Component Approach, 212 4.3 Statistics for Order Specification, 220 4.4 Finding Kronecker Indices, 222 4.5 Finding Scalar Component Models, 226 4.6 Estimation, 237 4.7 An Example, 245 4.8 Appendix: Canonical Correlation Analysis, 259 Exercises, 262 5 Unit-Root Nonstationary Processes 265 5.1 Univariate Unit-Root Processes, 266 5.2 Multivariate Unit-Root Processes, 279 5.3 Spurious Regressions, 290 5.4 Multivariate Exponential Smoothing, 291 5.5 Cointegration, 294 5.6 An Error-Correction Form, 297 5.7 Implications of Cointegrating Vectors, 300 5.8 Parameterization of Cointegrating Vectors, 302 5.9 Cointegration Tests, 303 5.10 Estimation of Error-Correction Models, 313 5.11 Applications, 319 5.12 Discussion, 326 5.13 Appendix, 327 Exercises, 328 6 Factor Models and Selected Topics 333 6.1 Seasonal Models, 333 6.2 Principal Component Analysis, 341 6.3 Use of Exogenous Variables, 345 6.4 Missing Values, 357 6.5 Factor Models, 364 6.6 Classification and Clustering Analysis, 386 Exercises, 394 7 Multivariate Volatility Models 399 7.1 Testing Conditional Heteroscedasticity, 401 7.2 Estimation of Multivariate Volatility Models, 407 7.3 Diagnostic Checks of Volatility Models, 409 7.4 Exponentially Weighted Moving Average, 414 7.5 BEKK Models, 417 7.6 Cholesky Decomposition and Volatility Modeling, 420 7.7 Dynamic Conditional Correlation Models, 428 7.8 Orthogonal Transformation, 434 7.9 Copula-Based Models, 443 7.10 Principal Volatility Components, 454 Exercises, 461 Appendix A Review of Mathematics and Statistics 465 A.1 Review of Vectors and Matrices, 465 A.2 Least-Squares Estimation, 477 A.3 Multivariate Normal Distributions, 478 A.4 Multivariate Student-t Distribution, 479 A.5 Wishart and Inverted Wishart Distributions, 480 A.6 Vector and Matrix Differentials, 481 Index 489

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Book SynopsisProvides a sound, intuitive understanding of the basic concepts students need as they pursue careers in business, economics, and the life and social sciences. In this book, the author applies real-world orientation to concepts, problem-solving approach, straight forward and concise writing style, and comprehensive exercise sets.Table of ContentsChapter 1: Functions, Graphs, and Limits1.1Functions1.2The Graph of a Function1.3Lines and Linear Functions1.4Functional Models1.5Limits1.6One-Sided Limits and ContinuityChapter 2: Differentiation: Basic Concepts2.1The Derivative2.2Techniques of Differentiation2.3Product and Quotient Rules; Higher-Order Derivatives2.4The Chain Rule2.5Marginal Analysis and Approximations Using Increments2.6Implicit Differentiation and Related RatesChapter 3: Additional Applications of the Derivative3.1 Increasing and Decreasing Functions; Relative Extrema3.2 Concavity and Points of Inflection3.3 Curve Sketching3.4 Optimization; Elasticity of Demand3.5 Additional Applied OptimizationChapter 4: Exponential and Logarithmic Functions4.1 Exponential Functions; Continuous Compounding4.2 Logarithmic Functions4.3 Differentiation of Exponential and Logarithmic Functions4.4 Additional Applications; Exponential ModelsChapter 5: Integration5.1 Indefinite Integration and Differential Equations5.2 Integration by Substitution5.3 The Definite Integral and the Fundamental Theorem of Calculus5.4 Applying Definite Integration: Distribution of Wealth and Average Value5.5 Additional Applications to Business and Economics5.6 Additional Applications to the Life and Social SciencesChapter 6: Additional Topics in Integration6.1 Integration by Parts; Integral Tables6.2 Numerical Integration6.3 Improper Integrals6.4 Introduction to Continuous ProbabilityChapter 7: Calculus of Several Variables7.1 Functions of Several Variables7.2 Partial Derivatives7.3 Optimizing Functions of Two Variables7.4 The Method of Least-Squares7.5 Constrained Optimization: The Method of Lagrange Multipliers7.6 Double IntegralsAppendix A: Algebra ReviewA.1 A Brief Review of AlgebraA.2 Factoring Polynomials and Solving Systems of EquationsA.3 Evaluating Limits with L’Hopital’s RuleA.4 The Summation Notation

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Book SynopsisStatistical science s first coordinated manual of methods for analyzing ordered categorical data, now fully revised and updated, continues to present applications and case studies in fields as diverse as sociology, public health, ecology, marketing, and pharmacy.Table of ContentsPreface. 1. Introduction. 1.1. Ordinal Categorical Scales. 1.2. Advantages of Using Ordinal Methods. 1.3. Ordinal Modeling Versus Ordinary Regession Analysis. 1.4. Organization of This Book. 2. Ordinal Probabilities, Scores, and Odds Ratios. 2.1. Probabilities and Scores for an Ordered Categorical Scale. 2.2. Ordinal Odds Ratios for Contingency Tables. 2.3. Confidence Intervals for Ordinal Association Measures. 2.4. Conditional Association in Three-Way Tables. 2.5. Category Choice for Ordinal Variables. Chapter Notes. Exercises. 3. Logistic Regression Models Using Cumulative Logits. 3.1. Types of Logits for An Ordinal Response. 3.2. Cumulative Logit Models. 3.3. Proportional Odds Models: Properties and Interpretations. 3.4. Fitting and Inference for Cumulative Logit Models. 3.5. Checking Cumulative Logit Models. 3.6. Cumulative Logit Models Without Proportional Odds. 3.7. Connections with Nonparametric Rank Methods. Chapter Notes. Exercises. 4. Other Ordinal Logistic Regression Models. 4.1. Adjacent-Categories Logit Models. 4.2. Continuation-Ratio Logit Models. 4.3. Stereotype Model: Multiplicative Paired-Category Logits. Chapter Notes. Exercises. 5. Other Ordinal Multinomial Response Models. 5.1. Cumulative Link Models. 5.2. Cumulative Probit Models. 5.3. Cumulative Log-Log Links: Proportional Hazards Modeling. 5.4. Modeling Location and Dispersion Effects. 5.5. Ordinal ROC Curve Estimation. 5.6. Mean Response Models. Chapter Notes. Exercises. 6. Modeling Ordinal Association Structure. 6.1. Ordinary Loglinear Modeling. 6.2. Loglinear Model of Linear-by-Linear Association. 6.3. Row or Column Effects Association Models. 6.4. Association Models for Multiway Tables. 6.5. Multiplicative Association and Correlation Models. 6.6. Modeling Global Odds Ratios and Other Associations. Chapter Notes. Exercises. 7. Non-Model-Based Analysis of Ordinal Association. 7.1. Concordance and Discordance Measures of Association. 7.2. Correlation Measures for Contingency Tables. 7.3. Non-Model-Based Inference for Ordinal Association Measures. 7.4. Comparing Singly Ordered Multinomials. 7.5. Order-Restricted Inference with Inequality Constraints. 7.6. Small-Sample Ordinal Tests of Independence. 7.7. Other Rank-Based Statistical Methods for Ordered Categories. Appendix: Standard Errors for Ordinal Measures. Chapter Notes. Exercises. 8. Matched-Pairs Data with Ordered Categories. 8.1. Comparing Marginal Distributions for Matched Pairs. 8.2. Models Comparing Matched Marginal Distributions. 8.3. Models for The Joint Distribution in A Square Table. 8.4. Comparing Marginal Distributions for Matched Sets. 8.5. Analyzing Rater Agreement on an Ordinal Scale. 8.6. Modeling Ordinal Paired Preferences. Chapter Notes. Exercises. 9. Clustered Ordinal Responses: Marginal Models. 9.1. Marginal Ordinal Modeling with Explanatory Variables. 9.2. Marginal Ordinal Modeling: GEE Methods. 9.3. Transitional Ordinal Modeling, Given the Past. Chapter Notes. Exercises. 10. Clustered Ordinal Responses: Random Effects Models. 10.1. Ordinal Generalized Linear Mixed Models. 10.2. Examples of Ordinal Random Intercept Models. 10.3. Models with Multiple Random Effects. 10.4. Multilevel (Hierarchical) Ordinal Models. 10.5. Comparing Random Effects Models and Marginal Models. Chapter Notes. Exercises. 11. Bayesian Inference for Ordinal Response Data. 11.1. Bayesian Approach to Statistical Inference. 11.2. Estimating Multinomial Parameters. 11.3. Bayesian Ordinal Regression Modeling. 11.4. Bayesian Ordinal Association Modeling. 11.5. Bayesian Ordinal Multivariate Regression Modeling. 11.6. Bayesian Versus Frequentist Approaches to Analyzing Ordinal Data. Chapter Notes. Exercises. Appendix Software for Analyzing Ordinal Categorical Data. Bibliography. Example Index. Subject Index.

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Book SynopsisThis book takes the reader on a journey through the world of college mathematics, focusing on some of the most important concepts and results in the theories of polynomials, linear algebra, real analysis, differential equations, coordinate geometry, trigonometry, elementary number theory, combinatorics, and probability. Preliminary material provides an overview of common methods of proof: argument by contradiction, mathematical induction, pigeonhole principle, ordered sets, and invariants. Each chapter systematically presents a single subject within which problems are clustered in each section according to the specific topic. The exposition is driven by nearly 1300 problems and examples chosen from numerous sources from around the world; many original contributions come from the authors. The source, author, and historical background are cited whenever possible. Complete solutions to all problems are given at the end of the book. This second edition includes new sections on quadratic polynomials, curves in the plane, quadratic fields, combinatorics of numbers, and graph theory, and added problems or theoretical expansion of sections on polynomials, matrices, abstract algebra, limits of sequences and functions, derivatives and their applications, Stokes' theorem, analytical geometry, combinatorial geometry, and counting strategies. Using the W.L. Putnam Mathematical Competition for undergraduates as an inspiring symbol to build an appropriate math background for graduate studies in pure or applied mathematics, the reader is eased into transitioning from problem-solving at the high school level to the university and beyond, that is, to mathematical research. This work may be used as a study guide for the Putnam exam, as a text for many different problem-solving courses, and as a source of problems for standard courses in undergraduate mathematics. Putnam and Beyond is organized for independent study by undergraduate and graduate students, as well as teachers and researchers in the physical sciences who wish to expand their mathematical horizons.Table of ContentsPreface to the Second Edition.- Preface to the First Edition.- A Study Guide.- 1. Methods of Proof.- 2. Algebra.- 3. Real Analysis.- 4. Geometry and Trigonometry.- 5. Number Theory.- 6. Combinatorics and Probability.- Solutions.- Index of Notation.- Index.

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Book SynopsisThis is part one of a two-volume introduction to real analysis and is intended for honours undergraduates who have already been exposed to calculus. The emphasis is on rigour and on foundations. The material starts at the very beginning--the construction of the number systems and set theory--then goes on to the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and finally to the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. There are also appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of twenty-five to thirty lectures each.The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.The fourth edition incorporates a large number of additional corrections reported since the release of the third edition, as well as some additional new exercises.

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Book SynopsisDieses vierfarbige Lehrbuch wendet sich an Studierende der Mathematik in Bachelor- und Lehramts-Studiengängen. Es bietet in einem Band ein lebendiges Bild der mathematischen Inhalte, die üblicherweise im ersten Studienjahr behandelt werden (und etliches mehr). Mathematik-Studierende finden wichtige Begriffe, Sätze und Beweise ausführlich und mit vielen Beispielen erklärt und werden an grundlegende Konzepte und Methoden herangeführt.Im Mittelpunkt stehen das Verständnis der mathematischen Zusammenhänge und des Aufbaus der Theorie sowie die Strukturen und Ideen wichtiger Sätze und Beweise. Es wird nicht nur ein in sich geschlossenes Theoriengebäude dargestellt, sondern auch verdeutlicht, wie es entsteht und wozu die Inhalte später benötigt werden.Herausragende Merkmale sind:- durchgängig vierfarbiges Layout mit mehr als 600 Abbildungen- prägnant formulierte Kerngedanken bilden die Abschnittsüberschriften- Selbsttests in kurzen Abständen ermöglichen Lernkontrollen während des Lesens- farbige Merkkästen heben das Wichtigste hervor- „Unter-der-Lupe“-Boxen zoomen in Beweise hinein, motivieren und erklären Details- „Hintergrund-und-Ausblick“-Boxen stellen Zusammenhänge zu anderen Gebieten und weiterführenden Themen her- Zusammenfassungen zu jedem Kapitel sowie Übersichtsboxen- mehr als 400 Verständnisfragen, Rechenaufgaben und Aufgaben zu Beweisen- deutsch-englisches Symbol- und Begriffsglossar Der inhaltliche Schwerpunkt liegt auf den Themen der Vorlesungen Analysis 1 und 2 sowie Linearer Algebra 1 und 2. Behandelt werden darüber hinaus Inhalte und Methodenkompetenzen, die vielerorts im ersten Studienjahr der Mathematikausbildung vermittelt werden.Hinweise, Lösungswege und Ergebnisse zu allen Aufgaben des Buchs stehen als PDF-Dateien auf http://sn.pub/extras in dem Ordner für das Werk Arens et al, „Mathematik“, Copyrightjahr 2018 zur Verfügung. Das Buch wird allen Studierenden der Mathematik vom Beginn des Studiums bis in höhere Semester hinein ein verlässlicher Begleiter sein.Für die 2. Auflage ist es vollständig durchgesehen, an zahlreichen Stellen didaktisch weiter verbessert und um einige Themen ergänzt worden.Stimme zur ersten Auflage:„Besonders gut gefallen mir die Übersichtlichkeit und die Verständlichkeit, besonders aber die Sichtbarmachung der Verbindung von Analysis und linearer Algebra, die in den Erstsemestervorlesungen oft zu kurz kommt.” Sylvia Prinz, Institut für Mathematikdidaktik, Universität zu KölnTable of ContentsVorwort.- 1 Was ist Mathematik und was tun Mathematiker?- 2 Logik, Mengen, Abbildungen − die Sprache der Mathematik.- 2.1 Junktoren und Quantoren.- 2.2 Grundbegriffe aus der Mengenlehre.- 2.3 Abbildungen.- 2.4 Relationen.- Zusammenfassung.- Aufgaben.- 3 Algebraische Strukturen − ein Blick hinter die Rechenregeln.- 3.1 Gruppen.- 3.2 Homomorphismen.- 3.3 Körper.- 3.4 Ringe.- Zusammenfassung.- Aufgaben.- 4 Zahlbereiche − Basis nicht nur der Analysis.- 4.1 Reelle Zahlen.- 4.2 Körperaxiome für die reellen Zahlen.- 4.3 Anordnungsaxiome für die reellen Zahlen.- 4.4 Ein Vollständigkeitsaxiom für die reellen Zahlen.- 4.5 Natürliche Zahlen und vollständige Induktion.- 4.6 Ganze Zahlen und rationale Zahlen.- 4.7 Komplexe Zahlen: Ihre Arithmetik und Geometrie.- Zusammenfassung.- Aufgaben.- 5 Lineare Gleichungssysteme − ein Tor zur linearen Algebra.- 5.1 Erste Lösungsversuche.- 5.2 Das Lösungsverfahren von Gauß und Jordan.- 5.3 Das Lösungskriterium und die Struktur der Lösung.- Zusammenfassung.- Aufgaben.- 6 Vektorräume − von Basen und Dimensionen.- 6.1 Der Vektorraumbegriff.- 6.2 Beispiele von Vektorräumen.- 6.3 Untervektorräume.- 6.4 Basis und Dimension.- 6.5 Summe und Durchschnitt von Untervektorräumen.- Zusammenfassung.- Aufgaben.- 7 Analytische Geometrie − Rechnen statt Zeichnen.- 7.1 Punkte und Vektoren im Anschauungsraum.- 7.2 Das Skalarprodukt im Anschauungsraum.- 7.3 Weitere Produkte von Vektoren im Anschauungsraum.- 7.4 Abstände zwischen Punkten, Geraden und Ebenen.- 7.5 Wechsel zwischen kartesischen Koordinatensystemen.- Zusammenfassung.- Aufgaben.- 8 Folgen − der Weg ins Unendliche.- 8.1 Der Begriff einer Folge.- 8.2 Konvergenz.- 8.3 Häufungspunkte und Cauchy-Folgen.- Zusammenfassung.- Aufgaben.- 9 Funktionen und Stetigkeit − ε trifft auf δ.- 9.1 Grundlegendes zu Funktionen.- 9.2 Beschränkte und monotone Funktionen.- 9.3 Grenzwerte für Funktionen und die Stetigkeit.- 9.4 Abgeschlossene, offene, kompakte Mengen.- 9.5 Stetige Funktionen mit kompaktem Definitionsbereich, Zwischenwertsatz.- Zusammenfassung.- Aufgaben.- 10 Reihen − Summieren bis zum Letzten.- 10.1 Motivation und Definition.- 10.2 Kriterien für Konvergenz.- 10.3 Absolute Konvergenz.- 10.4 Kriterien für absolute Konvergenz.- Zusammenfassung.- Aufgaben.- 11 Potenzreihen − Alleskönner unter den Funktionen.- 11.1 Definition und Grundlagen.- 11.2 Die Darstellung von Funktionen durch Potenzreihen.- 11.3 Die Exponentialfunktion.- 11.4 Trigonometrische Funktionen.- 11.5 Der Logarithmus.- Zusammenfassung.- Aufgaben.- 12 Lineare Abbildungen und Matrizen − Brücken zwischen Vektorräumen.- 12.1 Definition und Beispiele.- 12.2 Verknüpfungen von linearen Abbildungen.- 12.3 Kern, Bild und die Dimensionsformel.- 12.4 Darstellungsmatrizen.- 12.5 Das Produkt von Matrizen.- 12.6 Das Invertieren von Matrizen.- 12.7 Elementarmatrizen.- 12.8 Basistransformation.- 12.9 Der Dualraum.- Zusammenfassung.- Aufgaben.- <13 Determinanten − Kenngrößen von Matrizen.- 13.1 Die Definition der Determinante.- 13.2 Determinanten von Endomorphismen.- 13.3 Berechnung der Determinante.- 13.4 Anwendungen der Determinante.- Zusammenfassung.- Aufgaben.- 14 Normalformen − Diagonalisieren und Triangulieren.- 14.1 Diagonalisierbarkeit.- 14.2 Eigenwerte und Eigenvektoren.- 14.3 Berechnung der Eigenwerte und Eigenvektoren.- 14.4 Algebraische und geometrische Vielfachheit.- 14.5 Die Exponentialfunktion für Matrizen.- 14.6 Das Triangulieren von Endomorphismen.- 14.7 Die Jordan-Normalform.- 14.8 Die Berechnung einer Jordan-Normalform und Jordan-Basis.- Zusammenfassung.- Aufgaben.- 15 Differenzialrechnung − die Linearisierung von Funktionen.- 15.1 Die Ableitung.- 15.2 Differenziationsregeln.- 15.3 Der Mittelwertsatz.- 15.4 Verhalten differenzierbarer Funktionen.- 15.5 Taylorreihen.- Zusammenfassung.- Aufgaben.- 16 Integrale − von lokal zu global.- 16.1 Integration von Treppenfunktionen.- 16.2 Das Lebesgue-Integral.- 16.3 Stammfunktionen.- 16.4 Integrationstechniken.- 16.5 Integration über unbeschränkte Intervalle oder Funktionen.- 16.6 Parameterabhängige Integrale.- 16.7 Weitere Integrationsbegriffe.- Zusammenfassung.- Aufgaben.- 17 Euklidische und unitäre Vektorräume − orthogonales Diagonalisieren.- 17.1 Euklidische Vektorräume.- 17.2 Norm, Abstand, Winkel, Orthogonalität.- 17.3 Orthonormalbasen und orthogonale Komplemente.- 17.4 Unitäre Vektorräume.- 17.5 Orthogonale und unitäre Endomorphismen.- 17.6 Selbstadjungierte Endomorphismen.- 17.7 Normale Endomorphismen.- Zusammenfassung.- Aufgaben.- 18 Quadriken − vielseitig nutzbare Punktmengen.- 18.1 Symmetrische Bilinearformen.- 18.2 Hermitesche Sesquilinearformen.- 18.3 Quadriken und ihre Hauptachsentransformation.- 18.4 Die Singulärwertzerlegung.- 18.5 Die Pseudoinverse einer linearen Abbildung.- Zusammenfassung.- Aufgaben.- 19 Funktionenräume − Analysis und lineare Algebra Hand in Hand.- 19.1 Metrische Räume und ihre Topologie, normierte Räume.- 19.2 Konvergenz und Stetigkeit in metrischen Räumen.- 19.3 Kompaktheit.- 19.4 Zusammenhangsbegriffe.- 19.5 Vollständigkeit.- 19.6 Banach- und Hilberträume.- Zusammenfassung.- Aufgaben.- 20 Differenzialgleichungen − Funktionen sind gesucht.- 20.1 Begriffsbildungen.- 20.2 Elementare analytische Techniken.- 20.3 Existenz und Eindeutigkeit.- 20.4 Grundlegende numerische Verfahren.- Zusammenfassung.- Aufgaben .- 21 Funktionen mehrerer Variablen − Differenzieren im Raum.- 21.1 Einführung.- 21.2 Differenzierbarkeitsbegriffe: Totale und partielle Differenzierbarkeit.- 21.3 Differenziationsregeln.- 21.4 Mittelwertsätze und Schranksätze.- 21.5 Höhere partielle Ableitungen und der der Vertauschungssatz von H. A. Schwarz.- 21.6 Taylor-Formel und lokale Extrema.- 21.7 Der Lokale Umkehrsatz.- 21.8 Der Satz über implizite Funktionen.- Zusammenfassung.- Aufgaben.- 22 Gebietsintegrale − das Ausmessen von Mengen.- 22.1 Definition und Eigenschaften.- 22.2 Die Berechnung von Integralen.- 22.3 Die Transformationsformel.- 22.4 Wichtige Koordinatensysteme.- Zusammenfassung.- Aufgaben.- 23 Vektoranalysis − im Zentrum steht der Gauß'sche Satz.- 23.1 Kurven und Kurvenintegrale.- 23.2 Flächen und Flächenintegrale.- 23.3 Der Gauß’sche Satz.- Zusammenfassung.- Aufgaben.- 24 Optimierung − ein sehr generelles Problem.- 24.1 Lineare Optimierung.- 24.2 Das Simplex-Verfahren.- 24.3 Dualitätstheorie.- Zusammenfassung.- Aufgaben.- 25 Elementare Zahlentheorie − Teiler und Vielfache.- 25.1 Teilbarkeit.- 25.2 Der euklidische Algorithmus.- 25.3 Der Fundamentalsatz der Arithmetik.- 25.4 ggT und kgV.- 25.5 Zahlentheoretische Funktionen.- 25.6 Rechnen mit Kongruenzen.- Zusammenfassung.- Aufgaben.- 26 Elemente der diskreten Mathematik − die Kunst des Zählens.- 26.1 Einführung in die Graphentheorie.- 26.2 Einführung in die Kombinatorik.- 26.3 Erzeugende Funktionen.- Zusammenfassung.- Aufgaben.- Hinweise zu den Aufgaben.- Lösungen zu den Aufgaben.- Symbolglossar.- Index.

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Book SynopsisThis classic work has been a unique resource for thousands of mathematicians, scientists and engineers since its first appearance in 1902. Never out of print, its continuing value lies in its thorough and exhaustive treatment of special functions of mathematical physics and the analysis of differential equations from which they emerge. The book also is of historical value as it was the first book in English to introduce the then modern methods of complex analysis. This fifth edition preserves the style and content of the original, but it has been supplemented with more recent results and references where appropriate. All the formulas have been checked and many corrections made. A complete bibliographical search has been conducted to present the references in modern form for ease of use. A new foreword by Professor S.J. Patterson sketches the circumstances of the book''s genesis and explains the reasons for its longevity. A welcome addition to any mathematician''s bookshelf, this will aTrade Review'Generations of mathematicians have referred to W&W, as it has been affectionately called, for information that is difficult to locate elsewhere, in particular, on special functions. This improved new edition will enable future generations to access and learn from one of the great classical texts in the mathematical literature. My personal references to W&W are legion; the cover of my worn copy has long been separated from the text because of constant use.' Bruce C. Berndt, University of Illinois at Urbana-Champaign'Many of us who often use special functions revere the classics of complex analysis from the early 20th century. The names of Copson, MacRobert and Titchmarsh come to mind. However, the grandfather, indeed the overarching prototype, for most of these books is the one always referred to as "Whittaker and Watson." Fortunately for the world of mathematics, Victor Moll has presided over this wonderful fifth edition. Victor has provided an exceptionally valuable introduction that provides summaries of each chapter with ties to modern work. This new edition makes it easier for all to use the immense resources therein. Thank you, Victor! Thank you, Cambridge University Press.' George Andrews, The Pennsylvania State University'In many cases the coverage here is still the best or one of the best available, and is concise and all in one volume.' Allen Stenger, Mathematical Association of AmericaTable of ContentsForeword S. J. Patterson; Introduction; Part I. The Process of Analysis: 1. Complex numbers; 2. The theory of convergence; 3. Continuous functions and uniform convergence; 4. The theory of Riemann integration; 5. The fundamental properties of analytic functions – Taylor's, Laurent's and Liouville's theorems; 6. The theory of residues – application to the evaluation of definite integrals; 7. The expansion of functions in infinite series; 8. Asymptotic expansions and summable series; 9. Fourier series and trigonometric series; 10. Linear differential equations; 11. Integral equations; Part II. The Transcendental Functions: 12. The Gamma-function; 13. The zeta-function of Riemann; 14. The hypergeometric function; 15. Legendre functions; 16. The confluent hypergeometric function; 17. Bessel functions; 18. The equations of mathematical physics; 19. Mathieu functions; 20. Elliptic functions. General theorems and the Weierstrassian functions; 21. The theta-functions; 22. The Jacobian elliptic functions; 23. Ellipsoidal harmonics and Lamé's equation; Appendix. The elementary transcendental functions; References; Author index; Subject index.

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Book SynopsisApplied Calculus for Business, Economics, and the Social and Life Sciences, Expanded Edition provides a sound, intuitive understanding of the basic concepts students need as they pursue careers in business, economics, and the life and social sciences. Students achieve success using this text as a result of the author''s applied and real-world orientation to concepts, problem-solving approach, straight forward and concise writing style, and comprehensive exercise sets. More than 100,000 students worldwide have studied from this text!Table of ContentsChapter 1: Functions, Graphs, and Limits1.1 Functions1.2 The Graph of a Function1.3 Linear Functions1.4 Functional Models1.5 Limits1.6 One-Sided Limits and ContinuityChapter 2: Differentiation: Basic Concepts2.1 The Derivative2.2 Techniques of Differentiation2.3 Product and Quotient Rules; Higher-Order Derivatives2.4 The Chain Rule2.5 Marginal Analysis and Approximations Using Increments2.6 Implicit Differentiation and Related RatesChapter 3: Additional Applications of the Derivative3.1 Increasing and Decreasing Functions; Relative Extrema3.2 Concavity and Points of Inflection3.3 Curve Sketching3.4 Optimization; Elasticity of Demand3.5 Additional Applied OptimizationChapter 4: Exponential and Logarithmic Functions4.1 Exponential Functions; Continuous Compounding4.2 Logarithmic Functions4.3 Differentiation of Exponential and Logarithmic Functions4.4 Applications; Exponential ModelsChapter 5: Integration5.1 Indefinite Integration with Applications5.2 Integration by Substitution5.3 The Definite Integral and the Fundamental Theorem of Calculus5.4 Applying Definite Integration: Area Between Curves and Average Value5.5 Additional Applications to Business and Economics5.6 Additional Applications to the Life and Social SciencesChapter 6: Additional Topics in Integration6.1 Integration by Parts; Integral Tables6.2 Numerical Integration6.3 Improper IntegralsChapter 7: Calculus of Several Variables7.1 Functions of Several Variables7.2 Partial Derivatives7.3 Optimizing Functions of Two Variables7.4 The Method of Least-Squares7.5 Constrained Optimization: The Method of Lagrange Multipliers7.6 Double IntegralsChapter 8: Trigonometric Functions8.1 Angle Measurement; Trigonometric Functions8.2 Derivatives of Trigonometric Functions8.3 Integrals of Trigonometric FunctionsChapter 9: Differential Equations9.1 Introduction to Differential Equations9.2 First-Order Linear Differential Equations9.3 Additional Applications of Differential Equations9.4 Approximate Solutions of Differential Equations9.5 Difference Equations; The Cobweb ModelChapter 10: Probability and Calculus10.1 Continuous Probability Distributions10.2 Expected Value and Variance10.3 Normal DistributionsChapter 11: Infinite Series and Taylor Series Approximations11.1 Infinite Series; Geometric Series11.2 Tests for Convergence11.3 Functions as Power Series; Taylor SeriesAppendix A: Algebra ReviewA.1 A Brief Review of AlgebraA.2 Factoring Polynomials and Solving Systems of EquationsA.3 Evaluating Limits with L’Hopital’s RuleA.4 The Summation Notation

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Book Synopsis

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Book Synopsis

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Book Synopsis

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Book Synopsis

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Book Synopsis

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Book SynopsisFor courses in second-year calculus, linear calculus and differential equations. This text explores the standard problem-solving techniques of multivariable mathematics — integrating vector algebra ideas with multivariable calculus and differential equations. This text offers a full year of study and the flexibility to design various one-term and two-term courses.Table of Contents 1. Vectors. 2. Equations and Matrices. 3. Vector Spaces and Linearity. 4. Derivatives. 5. Differentiability. 6. Vector Differential Calculus. 7. Multiple Integration. 8. Integrals and Derivatives on Curves. 9. Vector Field Theory. 10. First Order Differential Equations. 11. Second-Order Equations. 12. Introduction to Systems. 13. Matrix Methods. 14. Infinite Series.

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Book SynopsisAbout our authors Mike Sullivan recently retired as Professor of Mathematics at Chicago State University, having taught there for more than 30 years. He received his PhD in mathematics from Illinois Institute of Technology. He is a native of Chicago's South Side and currently resides in Oak Lawn, Illinois. Mike has 4 children; the 2 oldest have degrees in mathematics and assisted in proofing, checking examples and exercises, and writing solutions manuals for this project. His son Mike Sullivan, III co-authored the Sullivan Graphing with Data Analysis series as well as this series. Mike has authored or co-authored more than 10 books. He owns a travel agency and splits his time between a condo in Naples, Florida and a home in Oak Lawn, where he enjoys gardening. Michael Sullivan, III has training in mathematics, statistics and economics, with a varied teaching background that includes 27 years of instruction in both high school and colTable of ContentsTable of Contents Foundations: A Prelude to Functions F.1 The Distance and Midpoint Formulas F.2 Graphs of Equations in Two Variables; Intercepts; Symmetry F.3 Lines F.4 Circles Chapter Project Functions and Their Graphs 1.1 Functions 1.2 The Graph of a Function 1.3 Properties of Functions 1.4 Library of Functions; Piecewise-defined Functions 1.5 Graphing Techniques: Transformations 1.6 Mathematical Models: Building Functions 1.7 Building Mathematical Models Using Variation Chapter Review Chapter Test Chapter Projects Linear and Quadratic Functions 2.1 Properties of Linear Functions and Linear Models 2.2 Building Linear Models from Data 2.3 Quadratic Functions and Their Zeros 2.4 Properties of Quadratic Functions 2.5 Inequalities Involving Quadratic Functions 2.6 Building Quadratic Models from Verbal Descriptions and from Data 2.7 Complex Zeros of a Quadratic Function 2.8 Equations and Inequalities Involving the Absolute Value Function Chapter Review Chapter Test Cumulative Review Chapter Projects Polynomial and Rational Functions 3.1 Polynomial Functions and Models 3.2 The Real Zeros of a Polynomial Function 3.3 Complex Zeros; Fundamental Theorem of Algebra 3.4 Properties of Rational Functions 3.5 The Graph of a Rational Function 3.6 Polynomial and Rational Inequalities Chapter Review Chapter Test Cumulative Review Chapter Projects Exponential and Logarithmic Functions 4.1 Composite Functions 4.2 One-to-One Functions; Inverse Functions 4.3 Exponential Functions 4.4 Logarithmic Functions 4.5 Properties of Logarithms 4.6 Logarithmic and Exponential Equations 4.7 Financial Models 4.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models 4.9 Building Exponential, Logarithmic, and Logistic Models from Data Chapter Review Chapter Test Cumulative Review Chapter Projects Trigonometric Functions 5.1 Angles and Their Measure 5.2 Trigonometric Functions: Unit Circle Approach 5.3 Properties of the Trigonometric Functions 5.4 Graphs of the Sine and Cosine Functions 5.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions 5.6 Phase Shift; Sinusoidal Curve Fitting Chapter Review Chapter Test Cumulative Review Chapter Projects Analytic Trigonometry 6.1 The Inverse Sine, Cosine, and Tangent Functions 6.2 The Inverse Trigonometric Functions (Continued) 6.3 Trigonometric Equations 6.4 Trigonometric Identities 6.5 Sum and Difference Formulas 6.6 Double-angle and Half-angle Formulas 6.7 Product-to-Sum and Sum-to-Product Formulas Chapter Review Chapter Test Cumulative Review Chapter Projects Applications of Trigonometric Functions 7.1 Right Triangle Trigonometry; Applications 7.2 The Law of Sines 7.3 The Law of Cosines 7.4 Area of a Triangle 7.5 Simple Harmonic Motion; Damped Motion; Combining Waves Chapter Review Chapter Test Cumulative Review Chapter Projects Polar Coordinates; Vectors 8.1 Polar Coordinates 8.2 Polar Equations and Graphs 8.3 The Complex Plane; De Moivre’s Theorem 8.4 Vectors 8.5 The Dot Product 8.6 Vectors in Space 8.7 The Cross Product Chapter Review Chapter Test Cumulative Review Chapter Projects Analytic Geometry 9.1 Conics 9.2 The Parabola 9.3 The Ellipse 9.4 The Hyperbola 9.5 Rotation of Axes; General Form of a Conic 9.6 Polar Equations of Conics 9.7 Plane Curves and Parametric Equations Chapter Review Chapter Test Cumulative Review Chapter Projects Systems of Equations and Inequalities 10.1 Systems of Linear Equations: Substitution and Elimination 10.2 Systems of Linear Equations: Matrices 10.3 Systems of Linear Equations: Determinants 10.4 Matrix Algebra 10.5 Partial Fraction Decomposition 10.6 Systems of Nonlinear Equations 10.7 Systems of Inequalities 10.8 Linear Programming Chapter Review Chapter Test Cumulative Review Chapter Projects Sequences; Induction; the Binomial Theorem 11.1 Sequences 11.2 Arithmetic Sequences 11.3 Geometric Sequences; Geometric Series 11.4 Mathematical Induction 11.5 The Binomial Theorem Chapter Review Chapter Test Cumulative Review Chapter Projects Counting and Probability 12.1 Counting 12.2 Permutations and Combinations 12.3 Probability Chapter Review Chapter Test Cumulative Review Chapter Projects A Preview of Calculus: The Limit, Derivative, and Integral of a Function 13.1 Finding Limits Using Tables and Graphs 13.2 Algebra Techniques for Finding Limits 13.3 One-sided Limits; Continuous Functions 13.4 The Tangent Problem; The Derivative 13.5 The Area Problem; The Integral Chapter Review Chapter Test Chapter Projects Appendix A: Review A.1 Algebra Essentials A.2 Geometry Essentials A.3 Polynomials A.4 Factoring Polynomials A.5 Synthetic Division A.6 Rational Expressions A.7 nth Roots; Rational Exponents A.8 Solving Equations A.9 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications A.10 Interval Notation; Solving Inequalities A.11 Complex Numbers Appendix B: Graphing Utilities B.1 The Viewing Rectangle B.2 Using a Graphing Utility to Graph Equations B.3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry B.4 Using a Graphing Utility to Solve Equations B.5 Square Screens B.6 Using a Graphing Utility to Graph Inequalities B.7 Using a Graphing Utility to Solve Systems of Linear Equations B.8 Using a Graphing Utility to Graph a Polar Equation B.9 Using a Graphing Utility to Graph Parametric Equations Answers Photo Credits Index

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Book SynopsisTable of ContentsTable of Contents Review R.1 Real Numbers R.2 Algebra Essentials R.3 Geometry Essentials R.4 Polynomials R.5 Factoring Polynomials R.6 Synthetic Division R.7 Rational Expressions R.8 nth Roots; Rational Exponents Equations and Inequalities 1.1 Linear Equations 1.2 Quadratic Equations 1.3 Complex Numbers; Quadratic Equations in the Complex Number System 1.4 Radical Equations; Equations Quadratic in Form; Factorable Equations 1.5 Solving Inequalities 1.6 Equations and Inequalities Involving Absolute Value 1.7 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications Chapter 1 Review, Test, and Projects Graphs 2.1 The Distance and Midpoint Formulas 2.2 Graphs of Equations in Two Variables; Intercepts; Symmetry 2.3 Lines 2.4 Circles 2.5 Variation Chapter 2 Review, Test, and Projects Functions and Their Graphs 3.1 Functions 3.2 The Graph of a Function 3.3 Properties of Functions 3.4 Library of Functions; Piecewise-defined Functions 3.5 Graphing Techniques: Transformations 3.6 Mathematical Models: Building Functions Chapter 3 Review, Test, and Projects Linear and Quadratic Functions 4.1 Properties of Linear Functions and Linear Models 4.2 Building Linear Models from Data 4.3 Quadratic Functions and Their Properties 4.4 Build Quadratic Models from Verbal Descriptions and from Data 4.5 Inequalities Involving Quadratic Functions Chapter 4 Review, Test, and Projects Polynomial and Rational Functions 5.1 Polynomial Functions 5.2 Graphing Polynomials Functions; Models 5.3 Properties of Rational Functions 5.4 The Graph of a Rational Function 5.5 Polynomial and Rational Inequalities 5.6 The Real Zeros of a Polynomial Function Chapter 5 Review, Test, and Projects Exponential and Logarithmic Functions 6.1 Composite Functions 6.2 One-to-One Functions; Inverse Functions 6.3 Exponential Functions 6.4 Logarithmic Functions 6.5 Properties of Logarithms 6.6 Logarithmic and Exponential Equations 6.7 Financial Models 6.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models 6.9 Building Exponential, Logarithmic, and Logistic Models from Data Chapter 6 Review, Test, and Projects Trigonometric Functions 7.1 Angles, Arc, Length, and Circular Motion 7.2 Right Triangle Trigonometry 7.3 Computing the Values of Trigonometric Functions of Acute Angles 7.4 Trigonometric Functions of Any Angle 7.5 Unit Circle Approach; Properties of the Trigonometric Functions 7.6 Graphs of the Sine and Cosine Functions 7.7 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions 7.8 Phase Shift; Sinusoidal Curve Fitting Chapter 7 Review, Test, and Projects Analytic Trigonometry 8.1 The Inverse Sine, Cosine, and Tangent Functions 8.2 The Inverse Trigonometric Functions (Continued) 8.3 Trigonometric Equations 8.4 Trigonometric Identities 8.5 Sum and Difference Formulas 8.6 Double-angle and Half-angle Formulas 8.7 Product-to-Sum and Sum-to-Product Formulas Chapter 8 Review, Test, and Projects Applications of Trigonometric Functions 9.1 Applications Involving Right Triangles 9.2 The Law of Sines 9.3 The Law of Cosines 9.4 Area of a Triangle 9.5 Simple Harmonic Motion; Damped Motion; Combining Waves Chapter 9 Review, Test, and Projects Polar Coordinates; Vectors 10.1 Polar Coordinates 10.2 Polar Equations and Graphs 10.3 The Complex Plane; De Moivre’s Theorem 10.4 Vectors 10.5 The Dot Product Chapter 10 Review, Test, and Projects Analytic Geometry 11.1 Conics 11.2 The Parabola 11.3 The Ellipse 11.4 The Hyperbola 11.5 Rotation of Axes; General Form of a Conic 11.6 Polar Equations of Conics 11.7 Plane Curves and Parametric Equations Chapter 11 Review, Test, and Projects Systems of Equations and Inequalities 12.1 Systems of Linear Equations: Substitution and Elimination 12.2 Systems of Linear Equations: Matrices 12.3 Systems of Linear Equations: Determinants 12.4 Matrix Algebra 12.5 Partial Fraction Decomposition 12.6 Systems of Nonlinear Equations 12.7 Systems of Inequalities 12.8 Linear Programming Chapter 12 Review, Test, and Projects Sequences; Induction; the Binomial Theorem 13.1 Sequences 13.2 Arithmetic Sequences 13.3 Geometric Sequences; Geometric Series 13.4 Mathematical Induction 13.5 The Binomial Theorem Chapter 13 Review, Test, and Projects Counting and Probability 14.1 Counting 14.2 Permutations and Combinations 14.3 Probability Chapter 14 Review, Test, and Projects Appendix: Graphing Utilities A.1 The Viewing Rectangle A.2 Using a Graphing Utility to Graph Equations A.3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry A.4 Using a Graphing Utility to Solve Equations A.5 Square Screens A.6 Using a Graphing Utility to Graph Inequalities A.7 Using a Graphing Utility to Solve Systems of Linear Equations A.8 Using a Graphing Utility to Graph a Polar Equation A.9 Using a Graphing Utility to Graph Parametric Equations Answers Credits Index

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Book SynopsisTable of ContentsPreface Prerequisite Skills Diagnostic Test R. Functions, Graphs, and Models R.1 Graphs and Equations R.2 Functions and Models R.3 Finding Domain and Range R.4 Slope and Linear Functions R.5 Nonlinear Functions and Models R.6 Exponential and Logarithmic Functions R.7 Mathematical Modeling and Curve Fitting Chapter Summary Chapter Review Exercises Chapter Test Extended Technology Application: Average Price of a Movie Ticket 1. Differentiation 1.1 Limits: A Numerical and Graphical Approach 1.2 Algebraic Limits and Continuity 1.3 Average Rates of Change 1.4 Differentiation Using Limits and Difference Quotients 1.5 Leibniz Notation and the Power and Sum—Difference Rules 1.6 The Product and Quotient Rules 1.7 The Chain Rule 1.8 Higher-Order Derivatives Chapter Summary Chapter Review Exercises Chapter Test Extended Technology Application: Path of a Baseball: The Tale of the Tape 2. Exponential and Logarithmic Functions 2.1 Exponential and Logarithmic Functions of the Natural Base, e 2.2 Derivatives of Exponential (Base-e) Functions 2.3 Derivatives of Natural Logarithmic Functions 2.4 Applications: Uninhibited and Limited Growth Models 2.5 Applications: Exponential Decay 2.6 The Derivatives of ax and logax Chapter Summary Chapter Review Exercises Chapter Test Extended Technology Application: The Business of Motion Picture Revenue and DVD Release 3. Applications of Differentiation 3.1 Using First Derivatives to Classify Maximum and Minimum Values and Sketch Graphs 3.2 Using Second Derivatives to Classify Maximum and Minimum Values and Sketch Graphs 3.3 Graph Sketching: Asymptotes and Rational Functions 3.4 Optimization: Finding Absolute Maximum and Minimum Values 3.5 Optimization: Business, Economics, and General Applications 3.6 Marginals, Differentials, and Linearization 3.7 Elasticity of Demand 3.8 Implicit Differentiation and Logarithmic Differentiation 3.9 Related Rates Chapter Summary Chapter Review Exercises Chapter Test Extended Technology Application: Maximum Sustainable Harvest 4. Integration 4.1 Antidifferentiation 4.2 Antiderivatives as Areas 4.3 Area and Definite Integrals 4.4 Properties of Definite Integrals: Additive Property, Average Value, and Moving Average 4.5 Integration Techniques: Substitution 4.6 Integration Techniques: Integration by Parts 4.7 Numerical Integration Chapter Summary Chapter Review Exercises Chapter Test Extended Technology Application: Business and Economics: Distribution of Wealth 5. Applications of Integration 5.1 Consumer and Producer Surplus; Price Floors, Price Ceilings, and Deadweight Loss 5.2 Integrating Growth and Decay Models 5.3 Improper Integrals 5.4 Probability 5.5 Probability: Expected Value; the Normal Distribution 5.6 Volume 5.7 Differential Equations Chapter Summary Chapter Review Exercises Chapter Test Extended Technology Application: Curve Fitting and Volumes of Containers 6. Functions of Several Variables 6.1 Functions of Several Variables 6.2 Partial Derivatives 6.3 Maximum - Minimum Problems 6.4 An Application: The Least-Squares Technique 6.5 Constrained Optimization: Lagrange Multipliers and the Extreme-Value Theorem 6.6 Double Integrals Chapter Summary Chapter Review Exercises Chapter Test Extended Technology Application: Minimizing Employees’ Travel Time in a Building Cumulative Review Appendices: A: Review of Basic Algebra B: Indeterminate Forms and l’Hôpital’s Rule C: Regression and Microsoft Excel D: Areas for a Standard Normal Distribution E: Using Tables of Integration Formulas Answers Index of Applications Index

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Book SynopsisTable of Contents1. Graphs and Functions 1.1 The Distance and Midpoint Formulas 1.2 Graphs of Equations in Two Variables; Circles 1.3 Functions and Their Graphs 1.4 Properties of Functions 1.5 Library of Functions; Piecewise-defined Functions 1.6 Graphing Techniques: Transformations 1.7 One-to-One Functions; Inverse Functions Chapter 1 Review, Test, and Projects 2. Trigonometric Functions 2.1 Angles, Arc, Length, and Circular Motion 2.2 Trigonometric Functions: Unit Circle Approach 2.3 Properties of the Trigonometric Functions 2.4 Graphs of the Sine and Cosine Functions 2.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions 2.6 Phase Shift; Sinusoidal Curve Fitting Chapter 2 Review, Test, and Projects 3. Analytic Trigonometry 3.1 The Inverse Sine, Cosine, and Tangent Functions 3.2 The Inverse Trigonometric Functions (Continued) 3.3 Trigonometric Equations 3.4 Trigonometric Identities 3.5 Sum and Difference Formulas 3.6 Double-angle and Half-angle Formulas 3.7 Product-to-Sum and Sum-to-Product Formulas Chapter 3 Review, Test, and Projects 4. Applications of Trigonometric Functions 4.1 Right Triangle Trigonometry; Applications 4.2 The Law of Sines 4.3 The Law of Cosines 4.4 Area of a Triangle 4.5 Simple Harmonic Motion; Damped Motion; Combining Waves Chapter 4 Review, Test, and Projects 5. Polar Coordinates; Vectors 5.1 Polar Coordinates 5.2 Polar Equations and Graphs 5.3 The Complex Plane; De Moivre’s Theorem 5.4 Vectors 5.5 The Dot Product 5.6 Vectors in Space 5.7 The Cross Produc Chapter 5 Review, Test, and Projects 6. Analytic Geometry 6.1 Conics 6.2 The Parabola 6.3 The Ellipse 6.4 The Hyperbola 6.5 Rotation of Axes; General Form of a Conic 6.6 Polar Equations of Conics 6.7 Plane Curves and Parametric Equations Chapter 6 Review, Test, and Projects 7. Exponential and Logarithmic Functions 7.1 Exponential Functions 7.2 Logarithmic Functions 7.3 Properties of Logarithms 7.4 Logarithmic and Exponential Equations 7.5 Financial Models 7.6 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay 7.7 Building Exponential, Logarithmic, and Logistic Models from Data Chapter 7 Review, Test, and Projects Appendix A: Review A.1 Algebra Essentials A.2 Geometry Essentials A.3 Factoring Polynomials; Completing the Square A.4 Solving Equations A.5 Complex Numbers; Quadratic Equations in the Complex Number System A.6 Interval Notation; Solving Inequalities A.7 nth Roots; Rational Exponents A.8 Lines Appendix B: Graphing Utilities B.1 The Viewing Rectangle B.2 Using a Graphing Utility to Graph Equations B.3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry B.4 Using a Graphing Utility to Solve Equations B.5 Square Screens B.6 Using a Graphing Utility to Graph Inequalities B.7 Using a Graphing Utility to Solve Systems of Linear Equations B.8 Using a Graphing Utility to Graph a Polar Equation B.9 Using a Graphing Utility to Graph Parametric Equations

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Book SynopsisCalculus Set Free: Infinitesimals to the Rescue is a single-variable calculus textbook that incorporates the use of infinitesimal methods.Trade ReviewCalculus Set Free is a well-written and self-contained text which offers a novel and mathematically rigorous approach to the topics typically present in Calculus 1 and 2. The text is largely successful in what it sets out to accomplish, and teachers interested in offering an introduction to Calculus built on an alternative theoretical approach should consider this text. * John Ross, MAA Reviews *Table of ContentsReview 1: Hyperreals, Limits, and Continuity 2: Derivatives 3: Applications of the Derivative 4: Integration 5: Transcendental Functions 6: Applications of Integration 7: Techniques of Integration 8: Alternate Representations: Parametric and Polar Curves 9: Additional Applications of Integration 10: Sequences and Series

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Book SynopsisCalculus Set Free: Infinitesimals to the Rescue is a single-variable calculus textbook that incorporates the use of infinitesimal methods.Trade ReviewCalculus Set Free is a well-written and self-contained text which offers a novel and mathematically rigorous approach to the topics typically present in Calculus 1 and 2. The text is largely successful in what it sets out to accomplish, and teachers interested in offering an introduction to Calculus built on an alternative theoretical approach should consider this text. * John Ross, MAA Reviews *Table of ContentsReview 1: Hyperreals, Limits, and Continuity 2: Derivatives 3: Applications of the Derivative 4: Integration 5: Transcendental Functions 6: Applications of Integration 7: Techniques of Integration 8: Alternate Representations: Parametric and Polar Curves 9: Additional Applications of Integration 10: Sequences and Series

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Book SynopsisThis book provides a self-contained introduction to the mathematical theory of hyperbolic systems of conservation laws, with particular emphasis on the study of discontinuous solutions, characterized by the appearance of shock waves. This area has experienced substantial progress in very recent years thanks to the introduction of new techniques, in particular the front tracking algorithm and the semigroup approach. These techniques provide a solution to the long standing open problems of uniqueness and stability of entropy weak solutions. This monograph is the first to present a comprehensive account of these new, fundamental advances, mainly obtained by the author together with several collaborators. It also includes a detailed analysis of the stability and convergence of the front tracking algorithm. The book is addressed to graduate students as well as researchers. Both the elementary and the more advanced material are carefully explained, helping the reader''s visual intuition withTrade ReviewAn excellent and self-contained treatment of the mathematical theory of hyperbolic systems of conservation laws ... written in a clear and self-contained way and will be of great value for graduate students and specialists in the field. * EMS *

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Book SynopsisThis book is a new edition of Tensors and Manifolds: With Applications to Mechanics and Relativity which was published in 1992. It is based on courses taken by advanced undergraduate and beginning graduate students in mathematics and physics, giving an introduction to the expanse of modern mathematics and its application in modern physics. It aims to fill the gap between the basic courses and the highly technical and specialised courses which both mathematics and physics students require in their advanced training, while simultaneously trying to promote, at an early stage, a better appreciation and understanding of each other''s discipline. The book sets forth the basic principles of tensors and manifolds, describing how the mathematics underlies elegant geometrical models of classical mechanics, relativity and elementary particle physics. The existing material from the first edition has been reworked and extended in some sections to provide extra clarity, as well as additional problemTrade ReviewReview from previous edition Clearly written and self-contained and, in particular, the author has succeeded in combining mathematical rigor with a certain degree of informality in a satisfactory way. As such, this work will certainly be appreciated by a wide audience. * Mathematical Reviews, August 1993 *Table of Contents1. Vector spaces ; 2. Multilinear mappings and dual spaces ; 3. Tensor product spaces ; 4. Tensors ; 5. Symmetric and skew-symmetric tensors ; 6. Exterior (Grassmann) algebra ; 7. The tangent map of real cartesian spaces ; 8. Topological spaces ; 9. Differentiable manifolds ; 10. Submanifolds ; 11. Vector fields, 1-forms and other tensor fields ; 12. Differentiation and integration of differential forms ; 13. The flow and the Lie derivative of a vector field ; 14. Integrability conditions for distributions and for pfaffian systems ; 15. Pseudo-Riemannian manifolds ; 16. Connection 1-forms ; 17. Connection on manifolds ; 18. Mechanics ; 19. Additional topics in mechanics ; 20. A spacetime ; 21. Some physics on Minkowski spacetime ; 22. Einstein spacetimes ; 23. Spacetimes near an isolated star ; 24. Nonempty spacetimes ; 25. Lie groups ; 26. Fiber bundles ; 27. Connections on fiber bundles ; 28. Gauge theory

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Book SynopsisAnalysis 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 initiatTrade ReviewThe clear, concise writing makes this book ideal for equipping undergraduates with a solid conceptual framework for approaching analysis rigorously and confidently. * V.K. Chellamuthu, CHOICE *Table of Contents1: Preliminaries 2: Limit of a sequence, an idea, a definition, a tool 3: Interlude: different kinds of numbers 4: Up and down - increasing and decreasing sequences 5: Sampling a sequence - subsequences 6: Special (or specially awkward) examples 7: Endless sums - a first look at series 8: Continuous functions - the domain thinks that the graph is unbroken 9: Limit of a function 10: Epsilontics and functions 11: Infinity and function limits 12: Differentiation - the slope of the graph 13: The Cauchy condition - sequences whose terms pack tightly together 14: More about series 15: Uniform continuity - continuity's global cousin 16: Differentiation - mean value theorems, power series 17: Riemann integration - area under a graph 18: The elementary functions revisited

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Book SynopsisVector and complex calculus are essential for applications to electromagnetism, fluid and solid mechanics, and the differential geometry of surfaces. Moving beyond the limits of standard multivariable calculus courses, this comprehensive textbook takes students from the geometry and algebra of vectors, through to the key concepts and tools of vector calculus. Topics explored include the differential geometry of curves and surfaces, curvilinear coordinates, ending with a study of the essential elements of the calculus of functions of one complex variable. Vector and Complex Calculus is richly illustrated to help students develop a solid visual understanding of the material, and the tools and concepts explored are foundational for upper-level engineering and physics courses. Each chapter includes a section of exercises which lead the student to practice key concepts and explore further interesting results.

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Book SynopsisThis volume describes the relationship between systems of linear inequalities and the geometry of convex polygons, examines solution sets for systems of linear inequalities in two and three unknowns (extension of the processes introduced to systems in any number of unknowns is quite simple), and examines questions of the consistency or inconsistency of such systems. Finally, it discusses the field of linear programming, one of the principal applications of the theory of systems of linear inequalities. A proof of the duality theorem of linear programming is presented in the last section.

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Book SynopsisA guide for any student of vector analysis, this text separates those relationships which are topologically invariant from those which are not. Based on the essentially geometric nature of the subject, this approach builds consistently on students' prior knowledge and geometrical intuition.

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Book SynopsisA new approach to the foundations of single variable calculus, based on the introductory course taught at CaltechTrade Review"The author’s stress on repeatable techniques . . . and the real numbers treated as infinite decimals results in a distinctive excursion through familiar territory.”—Nick Lord, The Mathematical Gazette"A very useful and constructive way to teach the subject."—Dominic Thorrington, IMA“Every science and engineering student takes calculus, but few learn the subject with depth and rigor. Calculus for Cranks addresses this gap head-on, introducing fundamental concepts in analysis that are valuable for all students – not just math majors.”—Carina Curto, Professor of Mathematics, Pennsylvania State University “Nets Katz has written a calculus textbook for students who don’t like being lied to. It will be essential for those who are constantly harassing their teachers with questions beginning with ‘why’ and ‘how.’”—Deane Yang, Professor of Mathematics, New York University “Calculus for Cranks unspools like a good novel! Katz deftly weaves abstraction and computation into a single narrative, with an entertaining set of exercises along the way.”—Amie Wilkinson, Professor of Mathematics, University of Chicago “Blending formal and informal insights, Katz pulls back the curtain on calculus, revealing its foundations, especially for those who think they’ve seen it before.”—Francis Su, author of Mathematics for Human Flourishing “Calculus for Cranks is a beautiful, rigorous, intuitive, introduction to real and complex analysis starting from logical reasoning and the number system. I recommend it highly for serious students.”—Wilhelm Schlag, Professor of Mathematics, Yale University

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Book Synopsis

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Book Synopsis

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Book SynopsisEver since the groundbreaking work of J.J. Kohn in the early 1960s, there has been a significant interaction between the theory of partial differential equations and the function theory of several complex variables. Partial Differential Equations and Complex Analysis explores the background and plumbs the depths of this symbiosis. The book is an excellent introduction to a variety of topics and presents many of the basic elements of linear partial differential equations in the context of how they are applied to the study of complex analysis. The author treats the Dirichlet and Neumann problems for elliptic equations and the related Schauder regularity theory, and examines how those results apply to the boundary regularity of biholomorphic mappings. He studies the ?-Neumann problem, then considers applications to the complex function theory of several variables and to the Bergman projection.Table of ContentsThe Dirichlet Problem in the Complex Plane Review of Fourier Analysis Pseudodifferential Operators Elliptic Operators Elliptic Boundary Value Problems A Degenerate Elliptic Boundary Value Problem The ?- Neumann Problem Applications of the ?- Neumann Problem The Local Solvability Issue and a Look Back.

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Book SynopsisIn the library at Trinity College, Cambridge in 1976, George Andrews of Pennsylvania State University discovered a sheaf of pages in the handwriting of Srinivasa Ramanujan. Soon designated as "Ramanujan’s Lost Notebook," it contains considerable material on mock theta functions and undoubtedly dates from the last year of Ramanujan’s life.Trade Reviewhematicians interested in the work of Ramanujan, will delight in studying this book … ." (Andrew V. Sills, Mathematical Reviews, Issue 2005 m)Table of ContentsPreface.- Introduction.- The Rogers–Ramanujan Continued Fraction and Its Modular Properties.- Explicit Evaluations of the Rogers–Ramanujan Continued Fraction.- A Fragment on the Rogers–Ramanujan and Cubic Continued Fractions.- The Rogers–Ramanujan Continued Fraction and Its Connections with Partitions and Lambert Series.- Finite Rogers–Ramanujan Continued Fractions.- Other q-continued Fractions.- Asymptotic Formulas for Continued Fractions.- Ramanujan’s Continued Fraction for (q2; q3)8/(q; q3)8.- The Rogers–Fine Identity.- An Empirical Study of the Rogers–Ramanujan Identities.- Rogers–Ramanujan–Slater Type Identities.- Partial Fractions.- Hadamard Products for Two q-Series.- Integrals of Theta-functions.- Incomplete Elliptic Integrals.- Infinite Integrals of q-Products.- Modular Equations in Ramanujan’s Lost Notebook.- Fragments on Lambert Series.- Location Guide.- Provenance.- References.- Index.

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Book SynopsisThe study of stable singularities is based on the now classical theories of Hassler Whitney, who determined the generic singularities (or lack of them) of Rn ~ Rm (m ~ 2n - 1) and R2 ~ R2, and Marston Morse, for mappings who studied these singularities for Rn ~ R.Table of ContentsI: Preliminaries on Manifolds.- §1. Manifolds.- §2. Differentiable Mappings and Submanifolds.- §3. Tangent Spaces.- §4. Partitions of Unity.- §5. Vector Bundles.- §6. Integration of Vector Fields.- II: Transversality.- §1. Sard’s Theorem.- §2. Jet Bundles.- §3. The Whitney C? Topology.- §4. Transversality.- §5. The Whitney Embedding Theorem.- §6. Morse Theory.- §7. The Tubular Neighborhood Theorem.- III: Stable Mappings.- §1. Stable and Infinitesimally Stable Mappings.- §2. Examples.- §3. Immersions with Normal Crossings.- §4. Submersions with Folds.- IV: The Malgrange Preparation Theorem.- §1. The Weierstrass Preparation Theorem.- §2. The Malgrange Preparation Theorem.- §3. The Generalized Malgrange Preparation Theorem.- V: Various Equivalent Notions of Stability.- §1. Another Formulation of Infinitesimal Stability.- §2. Stability Under Deformations.- §3. A Characterization of Trivial Deformations.- §4. Infinitesimal Stability => Stability.- §5. Local Transverse Stability.- §6. Transverse Stability.- §7. Summary.- VI: Classification of Singularities, Part I: The Thom-Boardman Invariants.- §1. The Sr Classification.- §2. The Whitney Theory for Generic Mappings between 2-Manifolds.- §3. The Intrinsic Derivative.- §4. The Sr,s Singularities.- §5. The Thom-Boardman Stratification.- §6. Stable Maps Are Not Dense.- VII: Classification of Singularities, Part II: The Local Ring of a Singularity.- §1. Introduction.- §2. Finite Mappings.- §3. Contact Classes and Morin Singularities.- §4. Canonical Forms for Morin Singularities.- §5. Umbilics.- §6. Stable Mappings in Low Dimensions.- §A. Lie Groups.- Symbol Index.

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Book Synopsis"This book presents a basic introduction to complex analysis in both an interesting and a rigorous manner. It contains enough material for a full year's course, and the choice of material treated is reasonably standard and should be satisfactory for most first courses in complex analysis.Trade Review"This book presents a basic introduction to complex analysis in both an interesting and a rigorous manner. It contains enough material for a full year's course, and the choice of material treated is reasonably standard and should be satisfactory for most first courses in complex analysis. The approach to each topic appears to be carefully thought out both as to mathematical treatment and pedagogical presentation, and the end result is a very satisfactory book for classroom use or self-study." --MathSciNetTable of ContentsI. The Complex Number System.- §1. The real numbers.- §2. The field of complex numbers.- §3. The complex plane.- §4. Polar representation and roots of complex numbers.- §5. Lines and half planes in the complex plane.- §6. The extended plane and its spherical representation.- II. Metric Spaces and the Topology of ?.- §1. Definition and examples of metric spaces.- §2. Connectedness.- §3. Sequences and completeness.- §4. Compactness.- §5. Continuity.- §6. Uniform convergence.- III. Elementary Properties and Examples of Analytic Functions.- §1. Power series.- §2. Analytic functions.- §3. Analytic functions as mapping, Möbius transformations.- IV. Complex Integration.- §1. Riemann-Stieltjes integrals.- §2. Power series representation of analytic functions.- §3. Zeros of an analytic function.- §4. The index of a closed curve.- §5. Cauchy’s Theorem and Integral Formula.- §6. The homotopic version of Cauchy’s Theorem and simple connectivity.- §7. Counting zeros; the Open Mapping Theorem.- §8. Goursat’s Theorem.- V. Singularities.- §1. Classification of singularities.- §2. Residues.- §3. The Argument Principle.- VI. The Maximum Modulus Theorem.- §1. The Maximum Principle.- §2. Schwarz’s Lemma.- §3. Convex functions and Hadamard’s Three Circles Theorem.- §4. Phragm>én-Lindel>üf Theorem.- VII. Compactness and Convergence in ihe Space of Analytic Functions.- §1. The space of continuous functions C(G, ?).- §2. Spaccs of analytic functions.- §3. Spaccs of meromorphic functions.- §4. The Riemann Mapping Theorem.- §5. Weierstrass Factorization Theorem.- §6. Factorization of the sine function.- $7. The gamma function.- §8. The Riemann zeta function.- VIII. Runge’s Theorem.- §1. Runge’s Theorem.- §2. Simple connectedness.- §3. Mittag-Leffler’s Theorem.- IX. Analytic Continuation and Riemann Surfaces.- §1. Schwarz Reflection Principle.- $2. Analytic Continuation Along A Path.- §3. Monodromy Theorem.- §4. Topological Spaces and Neighborhood Systems.- $5. The Sheaf of Germs of Analytic Functions on an Open Set.- $6. Analytic Manifolds.- §7. Covering spaccs.- X. Harmonic Functions.- §1. Basic Properties of harmonic functions.- §2. Harmonic functions on a disk.- §3. Subharmonic and superharmonic functions.- §4. The Dirichlet Problem.- §5. Green’s Functions.- XI. Entire Functions.- §1. Jensen’s Formula.- §2. The genus and order of an entire function.- §3. Hadamard Factorization Theorem.- XII. The Range of an Analytic Function.- §1. Bloch’s Theorem.- §2. The Little Picard Theorem.- §3. Schottky’s Theorem.- §4. The Great Picard Theorem.- Appendix A: Calculus for Complex Valued Functions on an Interval.- Appendix B: Suggestions for Further Study and Bibliographical Notes.- References.- List of Symbols.

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