Real analysis, real variables Books
Springer Basic Elements of Real Analysis
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£44.99
Springer Inside Calculus
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£44.99
Elsevier Science NonSelfAdjoint Boundary Eigenvalue Problems
£126.99
Springer Variational Principles in Topology Multidimensional Minimal Surface Theory 42 Mathematics and its Applications
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£85.49
Springer Oceanic Circulation Models Combining Data and Dynamics 284 Nato Science Series C
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Springer Quasiconformal Mappings and Sobolev Spaces 54 Mathematics and its Applications
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Springer Phase Portraits of Control Dynamical Systems 63 Mathematics and its Applications
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Springer Advances in Probability Distributions with Given Marginals Beyond the Copulas 67 Mathematics and Its Applications
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Springer Spline Functions and Multivariate Interpolations 248 Mathematics and Its Applications
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£85.49
Springer Nonstandard Methods of Analysis 291 Mathematics and Its Applications
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Springer Trigonometric Fourier Series and Their Conjugates 372 Mathematics and Its Applications
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Springer Advanced Topics in Difference Equations 404 Mathematics and Its Applications
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Springer The Theory of Cubature Formulas Mathematics Its Applications 415 Mathematics and Its Applications
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Springer Focal Boundary Value Problems for Differential and Difference Equations 436 Mathematics and Its Applications
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£123.49
Springer Applications of Point Set Theory in Real Analysis 429 Mathematics and Its Applications
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Springer Advanced Integration Theory
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£123.49
Springer Nonlinear Analysis Differential Equations and Control Proceedings of the NATO Advanced Ssudy Institute Universite De Montreal Canada July 27August 7 528 Nato Science Series C
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£85.49
Springer Nonlinear Analysis Differential Equations and Control Proceedings of the NATO Advanced Study Institute Universite De Montreal Canada July 27August 7 528 Nato Science Series C
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Springer Oscillation Theory for Difference and Functional Differential Equations
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Springer Optimality Conditions Abnormal and Degenerate Problems 526 Mathematics and Its Applications
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Springer New Developments in Singularity Theory Proceedings of the NATO Advanced Study Institute Cambridge 21 Nato Science Series II
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Springer New Developments in Singularity Theory
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Springer New York Concrete Functional Calculus Springer Monographs in Mathematics
Book SynopsisConcrete Functional Calculus focuses primarily on differentiability of some nonlinear operators on functions or pairs of functions.Trade ReviewFrom the reviews:“This monograph is a thorough and masterful work on non-linear analysis designed to be read and studied by graduate students and professional mathematical researchers. The overall perspective and choice of material is highly novel and original. … It is a unique account of some key areas of modern analysis which will surely turn out to be invaluable for many researchers in this and related areas.” (David Applebaum, The Mathematical Gazette, Vol. 98 (541), March, 2014)“The present monograph is quite extensive and interesting. It is divided into twelve chapters on different topics on Functional calculus and an appendix on non-atomic measure spaces. … The book has many historical comments and remarks which clarify the developments of the theory. It has also an extensive bibliography with 258 references. … will be very useful for all interested readers in Real-Functional Analysis and Probability.” (Francisco L. Hernandez, The European Mathematical Society, January, 2012)“The monograph under review aims at analyzing properties such as Hölder continuity, differentiability and analyticity of various types of nonlinear operators which arises in the study of differential and integral equations and in applications to problems of statistics and probability. … this is an interesting book which contains a lot of material.” (Massimo Lanza de Cristoforis, Mathematical Reviews, Issue 2012 e)Table of ContentsPreface.- 1 Introduction and Overview.- 2 Definitions and Basic Properties of Extended Riemann-Stieltjes integrals.- 3 Phi-variation and p-variation; Inequalities for Integrals.- 4 Banach Algebras.- 5 Derivatives and Analyticity in Normed Spaces.- 6 Nemytskii Operators on Function Spaces.- 7 Nemytskii Oerators on Lp Spaces.- 8 Two-Function Composition.- 9 Product Integration.- 10 Nonlinear Differential and Integral Equations.- 11 Fourier Series.- 12 Stochastic Processes and Phi-Variation.- Appendix Nonatomic Measure Spaces.- References.- Subject Index.- Author Index.- Index of Notation.
£123.49
Springer A First Course in Calculus
Book SynopsisOne Review of Basic Material.- I Numbers and Functions.- II Graphs and Curves.- Two Differentiation and Elementary Functions.- III The Derivative.- IV Sine and Cosine.- V The Mean Value Theorem.- VI Sketching Curves.- VII Inverse Functions.- VIII Exponents and Logarithms.- Three Integration.- IX Integration.- X Properties of the Integral.- XI Techniques of Integration.- XII Applications of Integration.- Four Taylor's Formula and Series.- XIII Taylor's Formula.- XIV Series.- Five Functions of Several Variables.- XV Vectors.- XVI Differentiation of Vectors.- XVII Functions of Several Variables.- XVIII The Chain Rule and the Gradient.- Answer.Table of ContentsI: Review of Basic Material. * Numbers and Functions. * Graphs and Curves. II: Differention and Elementary Functions. * The Derivative. * Sine and Cosine. * The Mean Value Theorem. * Sketching Curves. * Inverse Functions * Exponents and Logarithms. III: Integration. * Integration. * Properties of the Integral. * Techniques of Integration. * Applications of Integration. IV: Taylor's Formula and Series. * Taylor's Formula. * Series. Appendix. V: Functions of Several Variables. * Vectors. * Differention of Vectors. * Functions of Several Variables. * The Chain Rule and the Gradient.
£44.99
Springer Intermediate Real Analysis
Book SynopsisI Preliminaries.- II Functions.- III Real Sequences and Their Limits.- IV Infinite Series of Real Numbers.- V Limits of Functions.- VI Continuous Functions.- VII Derivatives.- VIII Convex Functions.- IX L'Hôpital's RuleTaylor's Theorem.- X The Complex Numbers. Trigonometric Sums. Infinite Products.- XI More on Series: Sequences and Series of Functions.- XII Sequences and Series of Functions II.- XIII The Riemann Integral I.- XIV The Riemann Integral II.- XV Improper Integrals. Elliptic Integrals and Functions.Table of ContentsI Preliminaries.- 1. Sets.- 2. The Set ? of Real Numbers.- 3. Some Inequalities.- 4. Interval Sets, Unions, Intersections, and Differences of Sets.- 5. The Non-negative Integers.- 6. The Integers.- 7. The Rational Numbers.- 8. Boundedness: The Axiom of Completeness.- 9. Archemedean Property.- 10. Euclid’s Theorem and Some of Its Consequences.- 11. Irrational Numbers.- 12. The Noncompleteness of the Rational Number System.- 13. Absolute Value.- II Functions.- 1. Cartesian Product.- 2. Functions.- 3. Sequences of Elements of a Set.- 4. General Sums and Products.- 5. Bernoulli’s and Related Inequalities.- 6. Factorials.- 7. Onto Functions, nth Root of a Positive Real Number.- 8. Polynomials. Certain Irrational Numbers.- 9. One-to-One Functions. Monotonic Functions.- 10. Composites of Functions. One-to-One Correspondences. Inverses of Functions.- 11. Rational Exponents.- 12. Some Inequalities.- III Real Sequences and Their Limits.- 1. Partially and Linearly Ordered Sets.- 2. The Extended Real Number System ?*.- 3. Limit Superior and Limit Inferior of Real Sequences.- 4. Limits of Real Sequences.- 5. The Real Number e.- 6. Criteria for Numbers To Be Limits Superior or Inferior of Real Sequences.- 7. Algebra of Limits: Sums and Differences of Sequences.- 8. Algebra of Limits: Products and Quotients of Sequences.- 9. L’Hôpital’s Theorem for Real Sequences.- 10. Criteria for the Convergence of Real Sequences.- IV Infinite Series of Real Numbers.- 1. Infinite Series of Real Numbers. Convergence and Divergence.- 2. Alternating Series.- 3. Series Whose Terms Are Nonnegative.- 4. Comparison Tests for Series Having Nonnegative Terms.- 5. Ratio and Root Tests.- 6. Kummer’s and Raabe’s Tests.- 7. The Product of Infinite Series.- 8. The Sine and Cosine Functions.- 9. Rearrangements of Infinite Series and Absolute Convergence.- 10. Real Exponents.- V Limits of Functions.- 1. Convex Set of Real Numbers.- 2. Some Real-Valued Functions of a Real Variable.- 3. Neighborhoods of a Point. Accumulation Point of a Set.- 4. Limits of Functions.- 5. One-Sided Limits.- 6. Theorems on Limits of Functions.- 7. Some Special Limits.- 8. P(x) as x ? ± ?, Where P is a Polynomial on ?.- 9. Two Theorems on Limits of Functions. Cauchy Criterion for Functions.- VI Continuous Functions.- 1. Definitions.- 2. One-Sided Continuity. Points of Discontinuity.- 3. Theorems on Local Continuity.- 4. The Intermediate-Value Theorem.- 5. The Natural Logarithm: Logs to Any Base.- 6. Bolzano—Weierstrass Theorem and Some Consequences.- 7. Open Sets in ?.- 8. Functions Continuous on Bounded Closed Sets.- 9. Monotonie Functions. Inverses of Functions.- 10. Inverses of the Hyperbolic Functions.- 11. Uniform Continuity.- VII Derivatives.- 1. The Derivative of a Function.- 2. Continuity and Differentiability. Extended Differentiability.- 3. Evaluating Derivatives. Chain Rule.- 4. Higher-Order Derivatives.- 5. Mean-Value Theorems.- 6. Some Consequences of the Mean-Value Theorems.- 7. Applications of the Mean-Value Theorem. Euler’s Constant.- 8. An Application of Rolle’s Theorem to Legendre Polynomials.- VIII Convex Functions.- 1. Geometric Terminology.- 2. Convexity and Differentiability.- 3. Inflection Points.- 4. Trigonometric Functions.- 5. Some Remarks on Differentiability.- 6. Inverses of Trigonometric Functions. Tschebyscheff Polynomials.- 7. Log Convexity.- IX L’Hôpital’s Rule—Taylor’s Theorem.- 1. Cauchy’s Mean-Value Theorem.- 2. An Application to Means and Sums of Order t.- 3. The O?0 Notation for Functions.- 4. Taylor’s Theorem of Order n.- 5. Taylor and Maclaurin Series.- 6. The Binomial Series.- 7. Tests for Maxima and Minima.- 8. The Gamma Function.- 9. Log-Convexity and the Functional Equation for ?.- X The Complex Numbers. Trigonometric Sums. Infinite Products.- 1. Introduction.- 2. The Complex Number System.- 3. Polar Form of a Complex Number.- 4. The Exponential Function on ?.- 5. nth Roots of a Complex Number. Trigonometric Functions on ?.- 6. Evaluation of Certain Trigonometric Sums.- 7. Convergence and Divergence of Infinite Products.- 8. Absolute Convergence of Infinite Products.- 9. Sine and Cosine as Infinite Products. Wallis’ Product. Stirling’s Formula.- 10. Some Special Limits. Stirling’s Formula.- 11. Evaluation of Certain Constants Associated with the Gamma Function.- XI More on Series: Sequences and Series of Functions.- 1. Introduction.- 2. Cauchy’s Condensation Test.- 3. Gauss’ Test.- 4. Pointwise and Uniform Convergence.- 5. Applications to Power Series.- 6. A Continuous But Nowhere Differentiable Function.- 7. The Weierstrass Approximation Theorem.- 8. Uniform Convergence and Differentiability.- 9. Application to Power Series.- 10. Analyticity in a Neighborhood of x0. Criteria for Real Analyticity.- XII Sequences and Series of Functions II.- 1. Arithmetic Operations with Power Series.- 2. Bernoulli Numbers.- 3. An Application of Bernoulli Numbers.- 4. Infinite Series of Analytic Functions.- 5. Abel’s Summation Formula and Some of Its Consequences.- 6. More Tests for Uniform Convergence.- XIII The Riemann Integral I.- 1. Darboux Integrals.- 2. Order Properties of the Darboux Integral.- 3. Algebraic Properties of the Darboux Integral.- 4. The Riemann Integral.- 5. Primitives.- 6. Fundamental Theorem of the Calculus.- 7. The Substitution Formula for Definite Integrals.- 8. Integration by Parts.- 9. Integration by the Method of Partial Fractions.- XIV The Riemann Integral II.- 1. Uniform Convergence and R-Integrals.- 2. Mean-Value Theorems for Integrals.- 3. Young’s Inequality and Some of Its Applications.- 4. Integral Form of the Remainder in Taylor’s Theorem.- 5. Sets of Measure Zero. The Cantor Set.- XV Improper Integrals. Elliptic Integrals and Functions.- 1. Introduction. Definitions.- 2. Comparison Tests for Convergence of Improper Integrals.- 3. Absolute and Conditional Convergence of Improper Integrals.- 4. Integral Representation of the Gamma Function.- 5. The Beta Function.- 6. Evaluation of ?0+? (sin x)/x dx.- 7. Integral Tests for Convergence of Series.- 8. Jacobian Elliptic Functions.- 9. Addition Formulas.- 10. The Uniqueness of the s, c, and d in Theorem 8.1.- 11. Extending the Definition of the Jacobi Elliptic Functions.- 12. Other Elliptic Functions and Integrals.
£71.24
Springer Real Analysis via Sequences and Series
Book SynopsisPreface.- 1. Introduction.- 2. Sequences.- 3. Series.- 4. Limits of Functions.- 5. Continuity.- 6. Differentiability.- 7. The Riemann Integral.- 8. Taylor Polynomials and Taylor Series.- 9. The Fixed Point Problem.- 10. Sequences of Functions.- Bibliography.- Index.Trade Review“The list of main topics covered is quite standard: sequences, series, limits, continuity, differentiation, Riemann integration, uniform convergence … . This is a well-written textbook with an abundance of worked examples and exercises that is intended for a first course in analysis with modest ambitions.” (Brian S. Thomson, Mathematical Reviews, March, 2016)“The authors … introduce sequences and series at the beginning and build the fundamental concepts of analysis from them. … it achieves the same goal of introducing students to mathematical rigor and basic concepts and results in real analysis. … Summing Up: Recommended. Upper-division undergraduates.” (D. Z. Spicer, Choice, Vol. 53 (5), January, 2016)“This textbook is based on the central idea that concepts such as continuity, differentiation and integration are approached via the concepts of sequences and series. … Most of the sections are followed by exercises. The textbook is recommended for a first course in mathematical analysis.” (Sorin Gheorghe Gal, zbMATH, Vol. 1325.26002, 2016)Table of ContentsPreface.- 1. Introduction.- 2. Sequences.- 3. Series.- 4. Limits of Functions.- 5. Continuity.- 6. Differentiability.- 7. The Riemann Integral.- 8. Taylor Polynomials and Taylor Series.- 9. The Fixed Point Problem.- 10. Sequences of Functions.- Bibliography.- Index.
£44.99
Springer Real Analysis via Sequences and Series
Book SynopsisPreface.- 1. Introduction.- 2. Sequences.- 3. Series.- 4. Limits of Functions.- 5. Continuity.- 6. Differentiability.- 7. The Riemann Integral.- 8. Taylor Polynomials and Taylor Series.- 9. The Fixed Point Problem.- 10. Sequences of Functions.- Bibliography.- Index.Trade Review“The list of main topics covered is quite standard: sequences, series, limits, continuity, differentiation, Riemann integration, uniform convergence … . This is a well-written textbook with an abundance of worked examples and exercises that is intended for a first course in analysis with modest ambitions.” (Brian S. Thomson, Mathematical Reviews, March, 2016)“The authors … introduce sequences and series at the beginning and build the fundamental concepts of analysis from them. … it achieves the same goal of introducing students to mathematical rigor and basic concepts and results in real analysis. … Summing Up: Recommended. Upper-division undergraduates.” (D. Z. Spicer, Choice, Vol. 53 (5), January, 2016)“This textbook is based on the central idea that concepts such as continuity, differentiation and integration are approached via the concepts of sequences and series. … Most of the sections are followed by exercises. The textbook is recommended for a first course in mathematical analysis.” (Sorin Gheorghe Gal, zbMATH, Vol. 1325.26002, 2016)Table of ContentsPreface.- 1. Introduction.- 2. Sequences.- 3. Series.- 4. Limits of Functions.- 5. Continuity.- 6. Differentiability.- 7. The Riemann Integral.- 8. Taylor Polynomials and Taylor Series.- 9. The Fixed Point Problem.- 10. Sequences of Functions.- Bibliography.- Index.
£33.74
Springer London Ltd Real Analysis
Book SynopsisReal Analysis is a comprehensive introduction to this core subject and is ideal for self-study or as a course textbook for first and second-year undergraduates. Combining an informal style with precision mathematics, the book covers all the key topics with fully worked examples and exercises with solutions. All the concepts and techniques are deployed in examples in the final chapter to provide the student with a thorough understanding of this challenging subject. This book offers a fresh approach to a core subject and manages to provide a gentle and clear introduction without sacrificing rigour or accuracy.Trade ReviewVol. 85 (504), 2001) "The book is a clear and structured introduction to real analysis. ... Fully worked out examples and exercises with solutions extend and illustrate the theory. Written in an easy-to-read style, combining informality and precision, the book is ideal for self-study or as a course textbook for first- and second-year undergraduates." (I. Rasa, Zentralblatt MATH, Vol. 969, 2001)Table of Contents1. Introductory Ideas.- 1.1 Foreword for the Student: Is Analysis Necessary?.- 1.2 The Concept of Number.- 1.3 The Language of Set Theory.- 1.4 Real Numbers.- 1.5 Induction.- 1.6 Inequalities.- 2. Sequences and Series.- 2.1 Sequences.- 2.2 Sums, Products and Quotients.- 2.3 Monotonie Sequences.- 2.4 Cauchy Sequences.- 2.5 Series.- 2.6 The Comparison Test.- 2.7 Series of Positive and Negative Terms.- 3. Functions and Continuity.- 3.1 Functions, Graphs.- 3.2 Sums, Products, Compositions; Polynomial and Rational Functions.- 3.3 Circular Functions.- 3.4 Limits.- 3.5 Continuity.- 3.6 Uniform Continuity.- 3.7 Inverse Functions.- 4. Differentiation.- 4.1 The Derivative.- 4.2 The Mean Value Theorems.- 4.3 Inverse Functions.- 4.4 Higher Derivatives.- 4.5 Taylor’s Theorem.- 5. Integration.- 5.1 The Riemann Integral.- 5.2 Classes of Integrable Functions.- 5.3 Properties of Integrals.- 5.4 The Fundamental Theorem.- 5.5 Techniques of Integration.- 5.6 Improper Integrals of the First Kind.- 5.7 Improper Integrals of the Second Kind.- 6. The Logarithmic and Exponential Functions.- 6.1 A Function Defined by an Integral.- 6.2 The Inverse Function.- 6.3 Further Properties of the Exponential and Logarithmic Functions.- Sequences and Series of Functions.- 7.1 Uniform Convergence.- 7.2 Uniform Convergence of Series.- 7.3 Power Series.- 8. The Circular Functions.- 8.1 Definitions and Elementary Properties.- 8.2 Length.- 9. Miscellaneous Examples.- 9.1 Wallis’s Formula.- 9.2 Stirling’s Formula.- 9.3 A Continuous, Nowhere Differentiable Function.- Solutions to Exercises.- The Greek Alphabet.
£24.95
Springer Nature Switzerland AG Theory of Infinite Sequences and Series
Book SynopsisThis textbook covers the majority of traditional topics of infinite sequences and series, starting from the very beginning – the definition and elementary properties of sequences of numbers, and ending with advanced results of uniform convergence and power series.The text is aimed at university students specializing in mathematics and natural sciences, and at all the readers interested in infinite sequences and series. It is designed for the reader who has a good working knowledge of calculus. No additional prior knowledge is required.The text is divided into five chapters, which can be grouped into two parts: the first two chapters are concerned with the sequences and series of numbers, while the remaining three chapters are devoted to the sequences and series of functions, including the power series. Within each major topic, the exposition is inductive and starts with rather simple definitions and/or examples, becoming more compressed and sophisticated as the course progresses. Each key notion and result is illustrated with examples explained in detail. Some more complicated topics and results are marked as complements and can be omitted on a first reading.The text includes a large number of problems and exercises, making it suitable for both classroom use and self-study. Many standard exercises are included in each section to develop basic techniques and test the understanding of key concepts. Other problems are more theoretically oriented and illustrate more intricate points of the theory, or provide counterexamples to false propositions which seem to be natural at first glance. Solutions to additional problems proposed at the end of each chapter are provided as an electronic supplement to this book.Trade Review“The text contains a large number of problems and exercises, which should make it suitable for both classroom use and self-study. Many standard exercises are included in each section to develop basic techniques and to test the understanding of concepts. … Many additional problems are proposed as homework tasks at the end of each chapter.” (Hüseyin Çakallı, zbMATH 1523.40001, 2023)Table of ContentsSequences of numbers.- Series of numbers.- Sequences of functions.- Series of functions.- Power series.
£54.99
Springer International Publishing AG Complex Integration: A Compendium of Smart and Little-Known Techniques for Evaluating Integrals and Sums
Book SynopsisIntegrals and sums are not generally considered for evaluation using complex integration. This book proposes techniques that mainly use complex integration and are quite different from those in the existing texts. Such techniques, ostensibly taught in Complex Analysis courses to undergraduate students who have had two semesters of calculus, are usually limited to a very small set of problems. Few practitioners consider complex integration as a tool for computing difficult integrals. While there are a number of books on the market that provide tutorials on this subject, the existing texts in this field focus on real methods. Accordingly, this book offers an eye-opening experience for computation enthusiasts used to relying on clever substitutions and transformations to evaluate integrals and sums. The book is the result of nine years of providing solutions to difficult calculus problems on forums such as Math Stack Exchange or the author's website, residuetheorem.com. It serves to detail to the enthusiastic mathematics undergraduate, or the physics or engineering graduate student, the art and science of evaluating difficult integrals, sums, and products.Table of Contents1. Review of foundational concepts1.1. Sequences and Series 1.1.1. Sequences of Real Numbers and their Series – sequences, limits, series, convergence, harmonic numbers, summation by parts, change in the order of summation 1.1.2. Power Series and Generating Functions – definitions, radius of convergence, generating function representations of sequences, convolution 1.2. Integrals 1.2.1. Riemann Sums – definition, direct evaluation of certain sums 1.2.2. Fundamental Theorem - definition of definite integral, statement of theorem, verifications 1.2.3. Multiple Integrals – double integrals, conditions for reversal or order of integration 1.3. Evaluation Techniques 1.3.1. Integration by Parts - review 1.3.2. Conversion to Multiple Integrals – “Feynman’s Technique,” replacing a portion of an integrand with an integral representation and reversing the order of integration 1.3.3. Green’s Theorem – review, path integrals and parametrization, Stokes’ Theorem, applications 1.3.4. Partial Fractions review 1.4. Problems 2. Complex Integration 2.1. Analytic Functions 2.1.1. Cauchy-Riemann Conditions – complex functions and their derivatives, defining analytic functions as a direction-independent derivative, harmonic functions 2.1.2. Evaluating Complex Integrals – numerical examples of parametrizations 2.1.3. Path Independence – demonstrate for analytic functions and demonstrate invalidity for nonanalytic integrands 2.2. Cauchy’s Theorems 2.2.1. Winding Numbers – definition in terms of a complex integral 2.2.2. Cauchy’s Integral Theorem – derivation and illustration for a wide variety of integrands and contours 2.2.3. Cauchy’s Theorem – statement, examples, Liouville’s Theorem, Morera’s Theorem 2.3. Useful Results 2.3.1. Taylor Series – review, error analysis in complex plane, convergence 2.3.2. Laurent Series – regions of validity (e.g., annuli), analytic continuation 2.3.3. Argument Principle – derivation for zeroes and poles 2.3.4. Rouche’s Theorem – derivation, illustration for determining poles within integration contours 2.4. Multivalued Functions – branch points, branch cuts, Riemann surfaces 2.5. Problems 3. Evaluation of Real Integrals and Sums 3.1. Preliminary Matters 3.1.1. Poles and Residue Theory – residue definition, residue computation 3.1.2. Essential Singularities – computation of residues of essential singularities 3.1.3. Branch Points – illustration of a unified approach to expressing an integral of a function in terms of its singularities 3.2. Definite Integrals 3.2.1. Integrands Having Both Poles and Branch Points – e.g., integrands featuring logs and exponents less than -1 3.2.2. Integrands Defined Over - insertion of one higher power of log(z) in the integrand, residue backpropagation 3.2.3. Integrands Having Rational Functions of Polynomials and Trigonometric Functions – integration over the unit circle, modifying the unit circle in the presence of singularities, replacing monomial with a branch point in constructing a contour integral 3.2.4. Alternative Contours: Wedges, Rectangles, and Others – reducing the number of singularities in a contour to simplify calculation 3.2.5. Integrands Having Algebraic Functions and the Residue At Infinity – whole new paradigm in evaluating definite integrals with finite limits of an integrand having branch points at the finite limits, defining the residue at infinity, branch point at infinity 3.3. Sums 3.3.1. Complex Integral Representations – selection of integrand and contour to produce sums, demonstration of convergence of complex integral as contour expands to infinity 3.3.2. Examples – rational summands, summands with trigonometric functions 3.4. Problems 4. Cauchy Principal Value 4.1. Integrands Having Poles On the Contour 4.1.1. Definition of a Cauchy Principal Value – definition as a limit, illustration with simple examples 4.1.2. Managing Divergent Terms of a Contour Integral – detailed illustrations of evaluating definite integrals via complex integrals having contributions with divergent terms that cancel 4.2. Analytic Signals and Hilbert Transforms – equivalence of Cauchy-Riemann equations and Hilbert transforms of real and imaginary parts of an analytic function, illustrations of analytic signals having harmonic real and imaginary parts, examples of deriving imaginary parts of analytic function from real part 4.3. Problems 5. Integral Transforms 5.1. Preliminary Matters 5.1.1. The Dirac Delta Function – derivation via self-transform in Hilbert transform integrals, review of properties 5.1.2. A General Discussion of Integral Transforms - integral transforms require a computable inverse to be of any use, conditions under which inverses exist, general format of integral transforms 5.2. The Fourier Transform 5.2.1. Definition and Plancherel’s Theorem – mean square error, and inner product spaces, the Fourier Transform as a Principal Value 5.2.2. Jordan’s Lemma – evaluating Fourier integrals using complex integration, convergence conditions 5.2.3. Parseval’s Theorem – statement, examples of integral evaluations, Fourier series and application of theorem to sums 5.2.4. Convolution Theorem – statement and derivation, applications 5.2.5. Analyticity of the Fourier Transform In the Complex Plane - theorem relating rates of convergence of Fourier transforms and their inverses in the complex plane, strips of convergence, causality 5.2.6. Poisson Sum Formula – derivation, application to computation of error function to machine precision anywhere in the complex plane 5.3. The Laplace Transform 5.3.1. Definition – extending the discussion of analyticity of the Fourier transform with an exponentially decaying kernel rather than an oscillatory kernel, derivation of inverse as an integral in the complex plane 5.3.2. Convolution Theorem – derivation, examples, application to computing certain classes of definite integrals 5.3.3. Inversion Via Complex Integration 5.3.3.1. Solutions to Ordinary Differential Equations and Rational Transforms – initial conditions, homogeneous and inhomogeneous equations, inversion via the residue theorem 5.3.3.2. Solutions to Partial Differential Equations and Multivalued Transforms – heat equation produces multivalued transforms, evaluation of inverse Laplace transforms to derive solutions 5.4. The Mellin Transform 5.4.1. Definition discussion of strip of convergence, inverse Mellin transform 5.4.2. Convolution Theorem – derivation; NB this will be used in the next chapter 5.4.3. Scaling – expression of scaled integrals in terms of residues 5.5. Problems 6. Asymptotic Analysis 6.1. Definitions 6.1.1. Big-O, Little-O, and The Squiggle – i.e., definitions of asymptotic equivalence in specific limits 6.1.2. Asymptotic Series – definition, properties, numerical calculations, summation acceleration techniques 6.2. Integration by Parts – development of asymptotic series; limitations 6.2.1. Euler-Maclurin Formula – derivation of asymptotic series using integration by parts, application to evaluation of sums 6.3. Watson’s Lemma and h-Transforms – asymptotic behavior of monotonic integrands, application of Mellin transforms in derivation 6.3.1. Application to Complex Integration – evaluation of integrals with branch points at infinity using h-transforms 6.4. Laplace’s Method – asymptotic behavior of nonmonotomic, nonoscillatory integrals 6.5. The Method of Steepest Descents – deriving asymptotic behavior of complex integrals, derive behavior of real integrals by using Cauchy’s Theorem 6.6. Problems
£27.99
Springer A Primer on Semiconvex Functions in General Potential Theories
Book SynopsisPart I. Semiconvex apparatus.- Chapter 1. Differentiability of convex functions.- Chapter 2. Semiconvex functions and upper contact jets.- Chapter 3. The lemmas of Jensen and Slodkowski.- Chapter 4. Semiconvex approximation of semicontinuous functions.- Part II. General potential-theoretic analysis.- Chapter 5. General potential theories.- Chapter 6. Duality and monotonicity in general potential theories.- Chapter 7. Basic tools in nonlinear potential theory.- Chapter 8. Semiconvex functions and subharmonics.- Chapter 9. Comparison principles.- Chapter 10. From Euclidean spaces to manifolds: a brief note.
£59.99
Springer International Publishing AG Real Mathematical Analysis
Book SynopsisBased on an honors course taught by the author at UC Berkeley, this introduction to undergraduate real analysis gives a different emphasis by stressing the importance of pictures and hard problems. Topics include: a natural construction of the real numbers, four-dimensional visualization, basic point-set topology, function spaces, multivariable calculus via differential forms (leading to a simple proof of the Brouwer Fixed Point Theorem), and a pictorial treatment of Lebesgue theory. Over 150 detailed illustrations elucidate abstract concepts and salient points in proofs. The exposition is informal and relaxed, with many helpful asides, examples, some jokes, and occasional comments from mathematicians, such as Littlewood, Dieudonné, and Osserman. This book thus succeeds in being more comprehensive, more comprehensible, and more enjoyable, than standard introductions to analysis.New to the second edition of Real Mathematical Analysis is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri’s Principle, and, in many cases, the ability to see an integral result from measure theory. The presentation includes Vitali’s Covering Lemma, density points — which are rarely treated in books at this level — and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject.Trade Review"This book, in its second edition, provides the basic concepts of real analysis. ... I strongly recommend it to everyone who wishes to study real mathematical analysis." (Catalin Barbu, zbMATH 1329.26003, 2016)Table of ContentsReal Numbers.- A Taste of Topology.- Functions of a Real Variable.- Function Spaces.- Multivariable Calculus.- Lebesgue Theory.
£38.94
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Analysis I: Convergence, Elementary functions
Book SynopsisFunctions in R and C, including the theory of Fourier series, Fourier integrals and part of that of holomorphic functions, form the focal topic of these two volumes. Based on a course given by the author to large audiences at Paris VII University for many years, the exposition proceeds somewhat nonlinearly, blending rigorous mathematics skilfully with didactical and historical considerations. It sets out to illustrate the variety of possible approaches to the main results, in order to initiate the reader to methods, the underlying reasoning, and fundamental ideas. It is suitable for both teaching and self-study. In his familiar, personal style, the author emphasizes ideas over calculations and, avoiding the condensed style frequently found in textbooks, explains these ideas without parsimony of words. The French edition in four volumes, published from 1998, has met with resounding success: the first two volumes are now available in English.Trade ReviewFrom the reviews of the original French edition: "... The content is quite classical ... [...] The treatment is less classical: precise although unpedantic (rather far from the definition-theorem-corollary-style), it contains many interesting commentaries of epistemological, pedagogical, historical and even political nature. [...] The author gives frequent interesting hints on recent developments of mathematics connected to the concepts which are introduced. The Introduction also contains comments that are very unusual in a book on mathematical analysis, going from pedagogy to critique of the French scientific-military-industrial complex, but the sequence of ideas is introduced in such a way that readers are less surprised than they might be.J. Mawhin in Zentralblatt Mathematik (1999) From the reviews: "Analysis I is the translation of the first volume of Godement’s four-volume work Analyse Mathématique, which offers a development of analysis more or less from the beginning up to some rather advanced topics. … the organization of the material is radically different … . It would … make excellent supplementary reading for honors calculus courses." (Gerald B. Folland, SIAM Review, Vol. 47 (3), 2005) "A book on analysis that is quite different from all other books on this subject. … for those who essentially know the material (the level of an average graduate student, say), and who are interested in mathematics will certainly love reading it. Those who lecture this material may find a lot of inspiration to make their lessons entertaining." (Adhemar Bultheel, Bulletin of the Belgian Mathematical Society, Vol. 12 (2), 2005) "Analysis I is an English translation of the first volume of a four-volume work. Analysis I consists of a spirally organized, organic, non-linear treatment of the introductory areas of ‘mathematical analysis as it was and as it has become’. It is infused with some excellent, sensitive appreciations of the work of pioneers … and reads as a heady blend of both classical concerns and modern refinements, often illuminated by a variety of approaches." (Nick Lord, The Mathematical Gazette, March, 2005)Table of ContentsI Sets and Functions.- §1. Set Theory.- 1 - Membership, equality, empty set.- 2 - The set defined by a relation. Intersections and unions.- 3 - Whole numbers. Infinite sets.- 4 - Ordered pairs, Cartesian products, sets of subsets.- 5 - Functions, maps, correspondences.- 6 - Injections, surjections, bijections.- 7 - Equipotent sets. Countable sets.- 8 - The different types of infinity.- 9 - Ordinals and cardinals.- §2. The logic of logicians.- II - Convergence: Discrete variables.- §1. Convergent sequences and series.- 0 - Introduction: what is a real number?.- 1 - Algebraic operations and the order relation: axioms of ?.- 2 - Inequalities and intervals.- 3 - Local or asymptotic properties.- 4 - The concept of limit. Continuity and differentiability.- 5 - Convergent sequences: definition and examples.- 6 - The language of series.- 7 - The marvels of the harmonic series.- 8 - Algebraic operations on limits.- §2. Absolutely convergent series.- 9 - Increasing sequences. Upper bound of a set of real numbers.- 10 - The function log x. Roots of a positive number.- 11 - What is an integral?.- 12 - Series with positive terms.- 13 - Alternating series.- 14 - Classical absolutely convergent series.- 15 - Unconditional convergence: general case.- 16 - Comparison relations. Criteria of Cauchy and d’Alembert.- 17 - Infinite limits.- 18 - Unconditional convergence: associativity.- §3. First concepts of analytic functions.- 19 - The Taylor series.- 20 - The principle of analytic continuation.- 21 - The function cot x and the series ?1/n2k.- 22 - Multiplication of series. Composition of analytic functions Formal series.- 23 - The elliptic functions of Weierstrass.- III - Convergence: Continuous variables.- §1. The intermediate value theorem.- 1 - Limit values of a function. Open and closed sets.- 2 - Continuous functions.- 3 - Right and left limits of a monotone function.- 4 - The intermediate value theorem.- §2. Uniform convergence.- 5 - Limits of continuous functions.- 6 - A slip up of Cauchy’s.- 7 - The uniform metric.- 8 - Series of continuous functions. Normal convergence.- §3. Bolzano-Weierstrass and Cauchy’s criterion.- 9 - Nested intervals, Bolzano-Weierstrass, compact sets.- 10 - Cauchy’s general convergence criterion.- 11 - Cauchy’s criterion for series: examples.- 12 - Limits of limits.- 13 - Passing to the limit in a series of functions.- §4. Differentiable functions.- 14 - Derivatives of a function.- 15 - Rules for calculating derivatives.- 16 - The mean value theorem.- 17 - Sequences and series of differentiable functions.- 18 - Extensions to unconditional convergence.- §5. Differentiable functions of several variables.- 19 - Partial derivatives and differentials.- 20 - Differentiability of functions of class C1.- 21 - Differentiation of composite functions.- 22 - Limits of differentiable functions.- 23 - Interchanging the order of differentiation.- 24 - Implicit functions.- Appendix to Chapter III.- 1 - Cartesian spaces and general metric spaces.- 2 - Open and closed sets.- 3 - Limits and Cauchy’s criterion in a metric space; complete spaces.- 4 - Continuous functions.- 5 - Absolutely convergent series in a Banach space.- 6 - Continuous linear maps.- 7 - Compact spaces.- 8 - Topological spaces.- IV Powers, Exponentials, Logarithms, Trigonometric Functions.- §1. Direct construction.- 1 - Rational exponents.- 2 - Definition of real powers.- 3 - The calculus of real exponents.- 4 - Logarithms to base a. Power functions.- 5 - Asymptotic behaviour.- 6 - Characterisations of the exponential, power and logarithmic functions.- 7 - Derivatives of the exponential functions: direct method.- 8 - Derivatives of exponential functions, powers and logarithms.- §2. Series expansions.- 9 - The number e. Napierian logarithms.- 10 - Exponential and logarithmic series: direct method.- 11 - Newton’s binomial series.- 12 - The power series for the logarithm.- 13 - The exponential function as a limit.- 14 - Imaginary exponentials and trigonometric functions.- 15 - Euler’s relation chez Euler.- 16 - Hyperbolic functions.- §3. Infinite products.- 17 - Absolutely convergent infinite products.- 18 - The infinite product for the sine function.- 19 - Expansion of an infinite product in series.- 20 - Strange identities.- §4. The topology of the functions Arg(z) and Log z.
£59.99
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Real Functions
Table of ContentsLocal systems.- Cluster sets.- Continuity.- Variation of a function.- Monotonicity.- Relations among derivates.- The denjoy-young relations.
£27.00
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Einführung in die Mathematik: Hintergründe der
Book Synopsis Diese Einführung besticht durch zwei ungewöhnliche Aspekte: Sie gibt einen Einblick in die Mathematik als Bestandteil unserer Kultur, und sie vermittelt die Hintergründe der Mathematik vom Schulstoff ausgehend bis zum Niveau von Mathematikvorlesungen im ersten Studienjahr. Die Stoffdarstellung geht vom Aufbau der natürlichen Zahlen aus; der Schwerpunkt liegt aber in den exakten Begründungen der Zahlenbegriffe, der Geometrie der Ebene und der Funktionen einer Veränderlichen. Dabei werden alle Sätze bis hin zum Hauptsatz der Algebra vollständig bewiesen. Der klare Aufbau des Buches mit Stichwortregister wichtiger Begriffe erleichtert das systematische Lernen und Nachschlagen. Die zweite Auflage enthält teilweise ausführliche Darstellungen für die Lösungen der zahlreichen Übungsaufgaben.Da viele Aspekte zur Sprache kommen, die so weder im Unterricht noch im Studium behandelt werden, ergänzt die Einführung ideal den Vorlesungsstoff für Lehramtskandidaten und Diplomstudenten.Trade Review"...dies ist eine Art "Brückenkurs"', der Aspekte der Schulmathematik von höherer Warte aus diskutiert... Der Autor steckt sich im Vorwort selbst das ehrgeizige Ziel, einen ‚Einblick in die Mathematik als einen Bestandteil unserer Kultur‘ zu geben, indem er sich ‚am Schulstoff (zwar) orientiert, aber über diesen hinausgeht und ihn hinterfragt.‘ Die Erreichbarkeit dieses Zieles stellt er mit diesem schönen Buch sehr überzeugend unter Beweis. Dabei wird beileibe nicht der Schulstoff ‚formalisiert‘, und noch weniger der Universitätsstoff ‚trivialisiert‘, sondern es kommen Aspekte zur Sprache, die im Mathematikunterricht wegen ihrer Schwierigkeit und im Mathematikstudium aus Zeitgründen kaum zur Sprache kommen. Dies ist ebenso verdienstvoll wie ungewöhnlich; als Ergebnis ist ein Buch herausgekommen, welches im ausufernden Markt tatsächlich eine Lücke füllt. Man kann grob drei Stoffgebiete unterscheiden, die behandelt werden, nämlich Zahlen (Kapitel 1-4 und 9), Geometrie (Kapitel 5 und 10) und Reelle Analysis (Kapitel 6-8). Wie ernst der Autor seine Aufgabe genommen hat, zeigt die sehr lesenswerte Einleitung, die auch den formalen Aufbau und inhaltliche Einzelheiten erklärt. Man kann allen Erstsemesterstudenten der Mathematik und Physik wärmstens empfehlen, dieses Buch als Ergänzung zu der von ihrem Dozenten empfohlenen Literatur zu kaufen und regelmäßig zu konsultieren." Jürgen Appell, Würzburg, in Zentralblatt MATH Table of ContentsNatürliche Zahlen.- Die 0 und die ganzen Zahlen.- Rationale Zahlen.- Reelle Zahlen.- Euklidische Geometrie der Ebene.- Reelle Funktionen einer Veränderlichen.- Maß und Integral.- Trigonometrie.- Die komplexen Zahlen.- Nicht-euklidische Geometrie.- Lösungen der Aufgaben.
£37.99
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Analysis II: Differential and Integral Calculus, Fourier Series, Holomorphic Functions
Book SynopsisFunctions in R and C, including the theory of Fourier series, Fourier integrals and part of that of holomorphic functions, form the focal topic of these two volumes. Based on a course given by the author to large audiences at Paris VII University for many years, the exposition proceeds somewhat nonlinearly, blending rigorous mathematics skilfully with didactical and historical considerations. It sets out to illustrate the variety of possible approaches to the main results, in order to initiate the reader to methods, the underlying reasoning, and fundamental ideas. It is suitable for both teaching and self-study. In his familiar, personal style, the author emphasizes ideas over calculations and, avoiding the condensed style frequently found in textbooks, explains these ideas without parsimony of words. The French edition in four volumes, published from 1998, has met with resounding success: the first two volumes are now available in English.Trade ReviewFrom the reviews of the original French edition: "... The content is quite classical ... [...] The treatment is less classical: precise although unpedantic (rather far from the definition-theorem-corollary-style), it contains many interesting commentaries of epistemological, pedagogical, historical and even political nature. [...] The author gives frequent interesting hints on recent developments of mathematics connected to the concepts which are introduced. The Introduction also contains comments that are very unusual in a book on mathematical analysis, going from pedagogy to critique of the French scientific-military-industrial complex, but the sequence of ideas is introduced in such a way that readers are less surprised than they might be.J. Mawhin in Zentralblatt Mathematik (1999) Table of ContentsDifferential and Integral Calculus.- The Riemann Integral.- Integrability Conditions.- The “Fundamental Theorem” (FT).- Integration by parts.- Taylor’s Formula.- The change of variable formula.- Generalised Riemann integrals.- Approximation Theorems.- Radon measures in ? or ?.- Schwartz distributions.- Asymptotic Analysis.- Truncated expansions.- Summation formulae.- Harmonic Analysis and Holomorphic Functions.- Analysis on the unit circle.- Elementary theorems on Fourier series.- Dirichlet’s method.- Analytic and holomorphic functions.- Harmonic functions and Fourier series.- From Fourier series to integrals.
£59.99
Elsevier Science Generalized Quantum Calculus with Applications
Book Synopsis
£79.96
Springer New York Infinite Processes Background To Analysis
Book Synopsisthirdly, as a kind of postscript to elementary analysis-as in a senior undergraduate course designed to reinforce students' understanding of elementary analysis and of elementary mathematics by considering the mathematical and historical connections between them.Table of ContentsI From Calculus to Analysis.- I.1 What’s Wrong with the Calculus?.- I.2 Growth and Change in Mathematics.- II Number.- II.1 Mathematics: Rational or Irrational?.- II.2 Constructive and Non-constructive Methods in Mathematics.- II.3 Common Measures, Highest Common Factors and the Game of Euclid.- II.4 Sides and Diagonals of Regular Polygons.- II.5 Numbers and Arithmetic—A Quick Review.- II.6 Infinite Decimals (Part 1).- II.7 Infinite Decimals (Part 2).- II.8 Recurring Nines.- II.9 Fractions and Recurring Decimals.- II.10 The Fundamental Property of Real Numbers.- II.11 The Arithmetic of Infinite Decimals.- II.12 Reflections on Recurring Themes.- II.13 Continued Fractions.- III Geometry.- III.1 Numbers and Geometry.- III.2 The Role of Geometrical Intuition.- III.3 Comparing Areas.- III.4 Comparing Volumes.- III.5 Curves and Surfaces.- IV Functions.- IV.1 What Is a Number?.- IV.2 What Is a Function?.- IV.3 What Is an Exponential Function?.
£42.74
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Probability in Banach Spaces: Isoperimetry and
Book SynopsisIsoperimetric, measure concentration and random process techniques appear at the basis of the modern understanding of Probability in Banach spaces. Based on these tools, the book presents a complete treatment of the main aspects of Probability in Banach spaces (integrability and limit theorems for vector valued random variables, boundedness and continuity of random processes) and of some of their links to Geometry of Banach spaces (via the type and cotype properties). Its purpose is to present some of the main aspects of this theory, from the foundations to the most important achievements. The main features of the investigation are the systematic use of isoperimetry and concentration of measure and abstract random process techniques (entropy and majorizing measures). Examples of these probabilistic tools and ideas to classical Banach space theory are further developed.Trade ReviewThis book gives an excellent, almost complete account of the whole subject of probability in Banach spaces, a branch of probability theory that has undergone vigorous development... There is no doubt in the reviewer's mind that this book [has] become a classic. MathSciNetAs the authors state, "this book tries to present some of the main aspects of the theory of probability in Banach spaces, from the foundation of the topic to the latest developments and current research questions''. The authors have succeeded admirably… This very comprehensive book develops a wide variety of the methods existing … in probability in Banach spaces. … It [has] become an event for mathematicians… Zentralblatt MATHTable of ContentsNotation.- 0. Isoperimetric Background and Generalities.- 1. Isoperimetric Inequalities and the Concentration of Measure Phenomenon.- 2. Generalities on Banach Space Valued Random Variables and Random Processes.- I. Banach Space Valued Random Variables and Their Strong Limiting Properties.- 3. Gaussian Random Variables.- 4. Rademacher Averages.- 5. Stable Random Variables.- 6 Sums of Independent Random Variables.- 7. The Strong Law of Large Numbers.- 8. The Law of the Iterated Logarithm.- II. Tightness of Vector Valued Random Variables and Regularity of Random Processes.- 9. Type and Cotype of Banach Spaces.- 10. The Central Limit Theorem.- 11. Regularity of Random Processes.- 12. Regularity of Gaussian and Stable Processes.- 13. Stationary Processes and Random Fourier Series.- 14. Empirical Process Methods in Probability in Banach Spaces.- 15. Applications to Banach Space Theory.- References.
£39.99
John Wiley & Sons Inc Introduction to Real Analysis
Book Synopsis* Helps one develop the ability to think deductively, analyse mathematical situations and extend ideas to a new context. * Maintains the same spirit and user-friendly approach with addition examples and expansion on Logical Operations and Set Theory.Table of ContentsCHAPTER 1 PRELIMINARIES. 1.1 Sets and Functions. 1.2 Mathematical Induction. 1.3 Finite and Infinite Sets. CHAPTER 2 THE REAL NUMBERS. 2.1 The Algebraic and Order Properties of R. 2.2 Absolute Value and the Real Line. 2.3 The Completeness Property of R. 2.4 Applications of the Supremum Property. 2.5 Intervals. CHAPTER 3 SEQUENCES AND SERIES. 3.1 Sequences and Their Limits. 3.2 Limit Theorems. 3.3 Monotone Sequences. 3.4 Subsequences and the Bolzano-Weierstrass Theorem. 3.5 The Cauchy Criterion. 3.6 Properly Divergent Sequences. 3.7 Introduction to Infinite Series. CHAPTER 4 LIMITS. 4.1 Limits of Functions. 4.2 Limit Theorems. 4.3 Some Extensions of the Limit Concept. CHAPTER 5 CONTINUOUS FUNCTIONS. 5.1 Continuous Functions. 5.2 Combinations of Continuous Functions. 5.3 Continuous Functions on Intervals. 5.4 Uniform Continuity. 5.5 Continuity and Gauges. 5.6 Monotone and Inverse Functions. CHAPTER 6 DIFFERENTIATION. 6.1 The Derivative. 6.2 The Mean Value Theorem. 6.3 L’Hospital’s Rules. 6.4 Taylor’s Theorem. CHAPTER 7 THE RIEMANN INTEGRAL. 7.1 Riemann Integral. 7.2 Riemann Integrable Functions. 7.3 The Fundamental Theorem. 7.4 The Darboux Integral. 7.5 Approximate Integration. CHAPTER 8 SEQUENCES OF FUNCTIONS. 8.1 Pointwise and Uniform Convergence. 8.2 Interchange of Limits. 8.3 The Exponential and Logarithmic Functions. 8.4 The Trigonometric Functions. CHAPTER 9 INFINITE SERIES. 9.1 Absolute Convergence. 9.2 Tests for Absolute Convergence. 9.3 Tests for Nonabsolute Convergence. 9.4 Series of Functions. CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL. 10.1 Definition and Main Properties. 10.2 Improper and Lebesgue Integrals. 10.3 Infinite Intervals. 10.4 Convergence Theorems. CHAPTER 11 A GLIMPSE INTO TOPOLOGY. 11.1 Open and Closed Sets in R. 11.2 Compact Sets. 11.3 Continuous Functions. 11.4 Metric Spaces. APPENDIX A LOGIC AND PROOFS. APPENDIX B FINITE AND COUNTABLE SETS. APPENDIX C THE RIEMANN AND LEBESGUE CRITERIA. APPENDIX D APPROXIMATE INTEGRATION. APPENDIX E TWO EXAMPLES. REFERENCES. PHOTO CREDITS. HINTS FOR SELECTED EXERCISES. INDEX.
£186.26
Springer-Verlag New York Inc. Measure Integral Derivative
Book SynopsisFeaturing over 180 exercises, this text for a one-semester course in Lebesgue's theory takes an elementary approach, making it easily accessible to both upper-undergraduate- and lower-graduate-level students.Trade ReviewFrom the reviews:“It is accessible to upper-undergraduate and lower graduate level students, and the only prerequisite is a course in elementary real analysis. … The book proposes 187 exercises where almost always the reader is proposed to prove a statement. … this book is a very helpful tool to get into Lebesgue’s theory in an easy manner.” (Daniel Cárdenas-Morales, zbMATH, Vol. 1277, 2014)“This is a brief … but enjoyable book on Lebesgue measure and Lebesgue integration at the advanced undergraduate level. … The presentation is clear, and detailed proofs of all results are given. … The book is certainly well suited for a one-semester undergraduate course in Lebesgue measure and Lebesgue integration. In addition, the long list of exercises provides the instructor with a useful collection of homework problems. Alternatively, the book could be used for self-study by the serious undergraduate student.” (Lars Olsen, Mathematical Reviews, December, 2013)Table of Contents1 Preliminaries.- 2 Lebesgue Measure.- 3 Lebesgue Integration.- 4 Differentiation and Integration.- A Measure and Integral over Unbounded Sets.- Index.
£46.74
Springer Elementary Analysis
Book SynopsisPreface.- 1 Introduction.- 2 Sequences.- 3 Continuity.- 4 Sequences and Series of Functions.- 5 Differentiation.- 6 Integration.- 7 Capstone.- Appendix on Set Notation.- Selected Hints and Answers.- References.- Index.Trade ReviewFrom the reviews of the first edition:"This book is intended for the student who has a good, but naïve, understanding of elementary calculus and now wishes to gain a thorough understanding of a few basic concepts in analysis, such as continuity, convergence of sequences and series of numbers, and convergence of sequences and series of functions. There are many nontrivial examples and exercises, which illuminate and extend the material. The author has tried to write in an informal but precise style, stressing motivation and methods of proof, and, in this reviewer’s opinion, has succeeded admirably."—MATHEMATICAL REVIEWS"This book occupies a niche between a calculus course and a full-blown real analysis course. … I think the book should be viewed as a text for a bridge or transition course that happens to be about analysis … . Lots of counterexamples. Most calculus books get the proof of the chain rule wrong, and Ross not only gives a correct proof but gives an example where the common mis-proof fails." —Allen Stenger (The Mathematical Association of America, June, 2008)Table of ContentsPreface.- 1 Introduction.- 2 Sequences.- 3 Continuity.- 4 Sequences and Series of Functions.- 5 Differentiation.- 6 Integration.- 7 Capstone.- Appendix on Set Notation.- Selected Hints and Answers.- References.- Index.
£34.84
Springer-Verlag New York Inc. Amazing and Aesthetic Aspects of Analysis
Book SynopsisLively prose and imaginative exercises draw the reader into this unique introductory real analysis textbook. Motivating the fundamental ideas and theorems that underpin real analysis with historical remarks and well-chosen quotes, the author shares his enthusiasm for the subject throughout. A student reading this book is invited not only to acquire proficiency in the fundamentals of analysis, but to develop an appreciation for abstraction and the language of its expression. In studying this book, students will encounter: the interconnections between set theory and mathematical statements and proofs; the fundamental axioms of the natural, integer, and real numbers; rigorous e-N and e-d definitions; convergence and properties of an infinite series, product, or continued fraction; series, product, and continued fraction formulæ for the various elementary functions and constants. ITrade Review Table of ContentsPreface.- Some of the most beautiful formulæ in the world.- Part 1. Some standard curriculum.- 1. Very naive set theory, functions, and proofs.- 2. Numbers, numbers, and more numbers.- 3. Infinite sequences of real and complex numbers.- 4. Limits, continuity, and elementary functions.- 5. Some of the most beautiful formulæ in the world I-III.- Part 2. Extracurricular activities.- 6. Advanced theory of infinite series.- 7. More on the infinite: Products and partial fractions.- 8. Infinite continued fractions.- Bibliography.- Index.
£55.99
Society for Industrial & Applied Mathematics,U.S. A Software Repository for Gaussian Quadratures
Book SynopsisThis companion piece to the author’s 2018 book, A Software Repository for Orthogonal Polynomials, focuses on Gaussian quadrature and the related Christoffel function. The book makes Gauss quadrature rules of any order easily accessible for a large variety of weight functions and for arbitrary precision. It also documents and illustrates known as well as original approximations for Gauss quadrature weights and Christoffel functions.The repository contains 60 datasets, each dealing with a particular weight function. Included are classical, quasi-classical, and, most of all, nonclassical weight functions and associated orthogonal polynomials.
£51.85
Society for Industrial & Applied Mathematics,U.S. The Classical Moment Problem and Some Related
Book SynopsisThe mathematical theory for many application areas depends on a deep understanding of the theory of moments. These areas include medical imaging, signal processing, computer visualization, and data science. The problem of moments has also found novel applications to areas such as control theory, image analysis, signal processing, polynomial optimization, and statistical big data. The Classical Moment Problem and Some Related Questions in Analysis presents: a unified treatment of the development of the classical moment problem from the late 19th century to the middle of the 20th century, important connections between the moment problem and many branches of analysis, a unified exposition of important classical results, which are difficult to read in the original journals, and a strong foundation for many areas in modern applied mathematics.
£60.35
Springer Nature Switzerland AG Convex Functions and Their Applications: A
Book SynopsisThorough introduction to an important area of mathematics Contains recent results Includes many exercisesTable of ContentsConvex Functions on Intervals.- Convex Sets in Real Linear Spaces.- Convex Functions on a Normed Linear Space.- Convexity and Majorization.- Convexity in Spaces of Matrices.- Duality and Convex Optimization.- Special Topics in Majorization Theory.- A. Generalized Convexity on Intervals.- B. Background on Convex Sets.- C. Elementary Symmetric Functions.- D. Second Order Differentiability of Convex Functions.- E. The Variational Approach of PDE.
£38.24
Springer Nature Switzerland AG Elements of Mathematics: A Problem-Centered
Book SynopsisThis textbook offers a rigorous presentation of mathematics before the advent of calculus. Fundamental concepts in algebra, geometry, and number theory are developed from the foundations of set theory along an elementary, inquiry-driven path. Thought-provoking examples and challenging problems inspired by mathematical contests motivate the theory, while frequent historical asides reveal the story of how the ideas were originally developed. Beginning with a thorough treatment of the natural numbers via Peano’s axioms, the opening chapters focus on establishing the natural, integral, rational, and real number systems. Plane geometry is introduced via Birkhoff’s axioms of metric geometry, and chapters on polynomials traverse arithmetical operations, roots, and factoring multivariate expressions. An elementary classification of conics is given, followed by an in-depth study of rational expressions. Exponential, logarithmic, and trigonometric functions complete the picture, driven by inequalities that compare them with polynomial and rational functions. Axioms and limits underpin the treatment throughout, offering not only powerful tools, but insights into non-trivial connections between topics. Elements of Mathematics is ideal for students seeking a deep and engaging mathematical challenge based on elementary tools. Whether enhancing the early undergraduate curriculum for high achievers, or constructing a reflective senior capstone, instructors will find ample material for enquiring mathematics majors. No formal prerequisites are assumed beyond high school algebra, making the book ideal for mathematics circles and competition preparation. Readers who are more advanced in their mathematical studies will appreciate the interleaving of ideas and illuminating historical details.Trade Review“Elements of mathematics is a curious book. The most challenging aspect of this volume to assess is its purpose.” (Jeff Johannes, Mathematical Reviews, October, 2022)“Transparency of explanation and gradually built material are outstanding features of the textbook. In addition, solutions to some problems are designed using more than one approach, making it adaptable to various students' backgrounds. … The book makes itself accessible to a vast population of students. The book can enhance the undergraduate curriculum or serve as a reflective resource for graduate mathematics students.” (Andrzej Sokolowski, MAA Reviews, March 20, 2022)“A historical concern is present throughout, with pieces of information on the history of concepts and theorems.” (Victor V. Pambuccian, zbMATH 1479.00002, 2022)Table of Contents0. Preliminaries: Sets, Relations, Maps.- 1. Natural, Integral and Rational Numbers.- 2. Real Numbers.- 3. Rational and Real Exponentiation.- 4. Limits of Real Functions.- 5. Real Analytic Plane Geometry.- 6. Polynomial Expressions.- 7. Polynomial Functions.- 8. Conics.- 9. Rational and Algebraic Expressions and Functions.- 10. Exponential and Logarithmic Functions.- 11. Trigonometry.- Further Reading.- Index.
£35.99
Springer Nature Switzerland AG New Perspectives on the Theory of Inequalities
Book SynopsisThis book provides new contributions to the theory of inequalities for integral and sum, and includes four chapters. In the first chapter, linear inequalities via interpolation polynomials and green functions are discussed. New results related to Popoviciu type linear inequalities via extension of the Montgomery identity, the Taylor formula, Abel-Gontscharoff's interpolation polynomials, Hermite interpolation polynomials and the Fink identity with Green’s functions, are presented. The second chapter is dedicated to Ostrowski’s inequality and results with applications to numerical integration and probability theory. The third chapter deals with results involving functions with nondecreasing increments. Real life applications are discussed, as well as and connection of functions with nondecreasing increments together with many important concepts including arithmetic integral mean, wright convex functions, convex functions, nabla-convex functions, Jensen m-convex functions, m-convex functions, m-nabla-convex functions, k-monotonic functions, absolutely monotonic functions, completely monotonic functions, Laplace transform and exponentially convex functions, by using the finite difference operator of order m. The fourth chapter is mainly based on Popoviciu and Cebysev-Popoviciu type identities and inequalities. In this last chapter, the authors present results by using delta and nabla operators of higher order.Trade Review“This is an interesting book on the theory of inequalities for integrals and sums, which researchers in this theory should have in their library.” (Gradimir Milovanović, Mathematical Reviews, December, 2023)Table of Contents1 Linear Inequalities via Interpolation Polynomials and Green Functions.- 2 Ostrowski Inequality.- 3 Functions with Nondecreasing Increments.- 4 Popoviciu and Cebysev-Popoviciu Type Identities and Inequalities.
£79.99
