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The derivative and the integral are the fundamental notions of calculus. Though there is essentially only one derivative, there is a variety of integrals, developed over the years for a variety of purposes, and this book describes them. No other single source treats all of the integrals of Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Steiltjes, Henstock-Kurzweil, Weiner, and Feynman. The basic properties of each are proved, their similarities and differences are pointed out, and the reason for their existence and their uses are given. There is plentiful historical information. The audience for the book is advanced undergraduate mathematics majors, graduate students, and faculty members. Even experienced faculty members are unlikely to be aware of all of the integrals in the Garden of Integrals and the book provides an opportunity to see them and appreciate their richness. Professor Burks clear and well-motivated exposition makes this book a joy to read. The book can serve as a reference, as a supplement to courses that include the theory of integration, and a source of exercises in analysis. There is no other book like it. ... Read more

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A beautiful book!
This is simply a beautiful exposition of the theory of integration. It lies somewhere in the space between a textbook and a monograph, developing the key results in each of a set of increasingly general theories of integration.

After motivating the contents of the book with a historical overview, beginning with a discussion of the computation of areas between curves in antiquity and moving on the development of the integral and differential calculi, Burk goes on to summarise the history of each of the modern integration theories he goes on to treat in the book. The key problems and personalities involved in the development of each theory are discussed, giving the book a nicely clear and focused historical flavour.

This is not to say that the book is not principally a technical exposition: that is precisely what it is! After providing us with the relevant historical particulars, Burk goes on to discuss the integration theories of Cauchy, Riemann, Lebesgue, and Henstock-Kurzweil; each theory subsumed in generality bt those that follow it. The conceptual difficulties that gave rise to each successive generalisation are discussed in quite a bit of detail, and a number of basic results - such as the Fundamental Theorem of Calculus, and the conditions under which a function is integrable - are proved for each theory. A liberal smattering of well chosen excercises aids the theoretical development, and help the reader to develop a good intuition about the nature and limitations of the various theories discussed.

The final two chapters are devoted to Weiner and the Feynman integrals, both of which are integrals of functionals. They are used respectively to model Brownian motion and to sum quantum mechanical probability amplitudes.

There is a genuine clarity and beauty in Burks's exposition, which makes this moderately technical book a pleasure to read. Whilst it is probably not appropriate as a primary text for students of introductory calculus(Burk draws liberally on the armamentarium of abstract analysis), parts of it could be profitably read by first-year undergraduates, and it would make for excellent auxiliary reading for students of analysis. (If you're a math grad who somehow managed to avoid learning about Lebesgue and Generalised Riemann integration as an undergrad, this is an excellent place to start)

For its historicity and the supreme clarity of its exposition, I would recommend this book to anyone who would like to extend the depth and clarity of his/her understanding of integration theory ... Read more

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From the Preface: ``The aim of this book is to offer a discussion of the fundamental concepts in the theory of differential and integral equations, with a view toward preparing the reader for graduate-level courses on advanced topics. This book can also serve a graduate-level course for non-mathematics majors. Students of various disciplines have found an increasing need to study such topics as control theory and systems theory, which are mainly based on differential equations, integral operators, and related concepts. The material included in this text will provide sufficient basis for understanding these areas of investigation. The prerequisites are a good course in advanced calculus ... some preparation in linear algebra, and a reasonable acquaintance with elementary complex analysis ... [T]he book can be used as a first course in differential and integral equations by any reader possessing knowledge of the above-mentioned fields.'' ... Read more

Customer Reviews (1)

Un excelente texto intermedio de ecuacuiones diferenciales
Tiene un buen nivel como texto intermedio en el área de las ecuaciones diferenciales,los capítulos 2 y 3 presentan los teoremas de existencia y unicidad de una manera clara y concisa. Otros dos capítulos muy buenos son los de ecuaciones integrales.Es recomendable para estudiantes que quieran profundizar en el tema. ... Read more

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This is a concise and elementary introduction to contemporary measure and integration theory as it is needed in many parts of analysis and probability theory. Undergraduate calculus and an introductory course on rigorous analysis in R are the only essential prerequisites, making the text suitable for both lecture courses and for self-study. Numerous illustrations and exercises are included to consolidate what has already been learned and to discover variants and extensions to the main material. Hints and solutions can be found on the authors website, which can be reached at http://www.motapa.de/measures_integrals_and_martingales/index.html ... Read more

Customer Reviews (2)

the best so far
This is yet another text in elementary measure theory and integration. What separates this text from the rest is an extremely clear presentation of the material plus some additional topics rarely found in other elementary textbooks. Explanations are clear and proofs very detailed. A nice introduction to martingales is included.
[...] Higly recommended to all grad students learning measure theory and integration.

nice self study guide
this can be a very good self-study guide for measure theory with little additional topics. As you can see in TOC there are numerous topics where some you can't find in "usual" measure theory books. The proof are detailed, much better than in other competing books, and clear. Generally, very easy to follow. With harder concepts the author prvides visual illustrations. Great help! The most useful is that the author provides solutions/hints for problems on the website. ... Read more

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/homepage/sac/cam/na2000/index.html7-Volume Set now available at special set price !

This volume contains contributions in the area of differential equations and integral equations.Many numerical methods have arisen in response to the need to solve "real-life" problems in applied mathematics, in particular problems that do not have a closed-form solution.Contributions on both initial-value problems and boundary-value problems in ordinary differential equations appear in this volume.Numerical methods for initial-value problems in ordinary differential equations fall naturally into two classes: those which use one starting value at each step (one-step methods) and those which are based on several values of the solution (multistep methods).

John Butcher has supplied an expert's perspective of the development of numerical methods for ordinary differential equations in the 20th century.

Rob Corless and Lawrence Shampine talk about established technology, namely software for initial-value problems using Runge-Kutta and Rosenbrock methods, with interpolants to fill in the solution between mesh-points, but the 'slant' is new - based on the question, "How should such software integrate into the current generation of Problem Solving Environments?"

Natalia Borovykh and Marc Spijker study the problem of establishing upper bounds for the norm of the nth power of square matrices.

The dynamical system viewpoint has been of great benefit to ODE theory and numerical methods.Related is the study of chaotic behaviour.

Willy Govaerts discusses the numerical methods for the computation and continuation of equilibria and bifurcation points of equilibria of dynamical systems.

Arieh Iserles and Antonella Zanna survey the construction of Runge-Kutta methods which preserve algebraic invariant functions.

Valeria Antohe and Ian Gladwell present numerical experiments on solving a Hamiltonian system of Hénon and Heiles with a symplectic and a nonsymplectic method with a variety of precisions and initial conditions.

Stiff differential equations first became recognized as special during the 1950s. In 1963 two seminal publications laid to the foundations for later development: Dahlquist's paper on A-stable multistep methods and Butcher's first paper on implicit Runge-Kutta methods.

Ernst Hairer and Gerhard Wanner deliver a survey which retraces the discovery of the order stars as well as the principal achievements obtained by that theory.

Guido Vanden Berghe, Hans De Meyer, Marnix Van Daele and Tanja Van Hecke construct exponentially fitted Runge-Kutta methods with s stages.

Differential-algebraic equations arise in control, in modelling of mechanical systems and in many other fields.

Jeff Cash describes a fairly recent class of formulae for the numerical solution of initial-value problems for stiff and differential-algebraic systems.

Shengtai Li and Linda Petzold describe methods and software for sensitivity analysis of solutions of DAE initial-value problems.

Again in the area of differential-algebraic systems, Neil Biehn, John Betts, Stephen Campbell and William Huffman present current work on mesh adaptation for DAE two-point boundary-value problems.

Contrasting approaches to the question of how good an approximation is as a solution of a given equation involve (i) attempting to estimate the actual error (i.e., the difference between the true and the approximate solutions) and (ii) attempting to estimate the defect - the amount by which the approximation fails to satisfy the given equation and any side-conditions.

The paper by Wayne Enright on defect control relates to carefully analyzed techniques that have been proposed both for ordinary differential equations and for delay differential equations in which an attempt is made to control an estimate of the size of the defect.

Many phenomena incorporate noise, and the numerical solution of stochastic differential equations has developed as a relatively new item of study in the area.

Keven Burrage, Pamela Burrage and Taketomo Mitsui review the way numerical methods for solving stochastic differential equations (SDE's) are constructed.

One of the more recent areas to attract scrutiny has been the area of differential equations with after-effect (retarded, delay, or neutral delay differential equations) and in this volume we include a number of papers on evolutionary problems in this area.

The paper of Genna Bocharov and Fathalla Rihan conveys the importance in mathematical biology of models using retarded differential equations.

The contribution by Christopher Baker is intended to convey much of the background necessary for the application of numerical methods and includes some original results on stability and on the solution of approximating equations.

Alfredo Bellen, Nicola Guglielmi and Marino Zennaro contribute to the analysis of stability of numerical solutions of nonlinear neutral differential equations.

Koen Engelborghs, Tatyana Luzyanina, Dirk Roose, Neville Ford and Volker Wulf consider the numerics of bifurcation in delay differential equations.

Evelyn Buckwar contributes a paper indicating the construction and analysis of a numerical strategy for stochastic delay differential equations (SDDEs).

This volume contains contributions on both Volterra and Fredholm-type integral equations.

Christopher Baker responded to a late challenge to craft a review of the theory of the basic numerics of Volterra integral and integro-differential equations.

Simon Shaw and John Whiteman discuss Galerkin methods for a type of Volterra integral equation that arises in modelling viscoelasticity.

A subclass of boundary-value problems for ordinary differential equation comprises eigenvalue problems such as Sturm-Liouville problems (SLP) and Schrödinger equations.

Liviu Ixaru describes the advances made over the last three decades in the field of piecewise perturbation methods for the numerical solution of Sturm-Liouville problems in general and systems of Schrödinger equations in particular.

Alan Andrew surveys the asymptotic correction method for regular Sturm-Liouville problems.

Leon Greenberg and Marco Marletta survey methods for higher-order Sturm-Liouville problems.

R. Moore in the 1960s first showed the feasibility of validated solutions of differential equations, that is, of computing guaranteed enclosures of solutions.

Boundary integral equations.Numerical solution of integral equations associated with boundary-value problems has experienced continuing interest.

Peter Junghanns and Bernd Silbermann present a selection of modern results concerning the numerical analysis of one-dimensional Cauchy singular integral equations, in particular the stability of operator sequences associated with different projection methods.

Johannes Elschner and Ivan Graham summarize the most important results achieved in the last years about the numerical solution of one-dimensional integral equations of Mellin type of means of projection methods and, in particular, by collocation methods.

A survey of results on quadrature methods for solving boundary integral equations is presented by Andreas Rathsfeld.

Wolfgang Hackbusch and Boris Khoromski present a novel approach for a very efficient treatment of integral operators.

Ernst Stephan examines multilevel methods for the h-, p- and hp- versions of the boundary element method, including pre-conditioning techniques.

George Hsiao, Olaf Steinbach and Wolfgang Wendland analyze various boundary element methods employed in local discretization schemes.
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This volume is a useful introduction to the subject of Fourier Integral Operators and is based on the author’s classic set of notes. Covering a range of topics from Hörmander’s exposition of the theory, Duistermaat approaches the subject from symplectic geometry and includes application to hyperbolic equations (= equations of wave type) and oscillatory asymptotic solutions which may have caustics. This text is suitable for mathematicians and (theoretical) physicists with an interest in (linear) partial differential equations, especially in wave propagation, rep. WKB-methods.

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This is the first book in world literature giving a systematic development of a general asymptotic theory for nonlinear partial differential equations with dissipation. Many typical well-known equations are considered as examples, such as: nonlinear heat equation, KdVB equation, nonlinear damped wave equation, Landau-Ginzburg equation, Sobolev type equations, systems of equations of Boussinesq, Navier-Stokes and others.

Product Description
The first part of this volume is a survey of the current state of the theory of linear integral equations. Material from abstract operator theory is presented first, before moving on to a detailed discussion of Fredholm operators and estimates for the eigenvalues and singular numbers of integral operators. Other sections of this part are devoted to the theories of one-, and higher-dimensional singular operators. The second part of the volume concentrates on theoretical aspects of the boundary integral equation method, with particular emphasis on the results for equations on non-smooth surfaces obtained during the last few years. A survey of the classical theory of harmonic potentials and the boundary integral equations on smooth surfaces generated by such potentials is given and the theory is then extended to elastic and hydrodynamic potentials and to other problems of mathematical physics, including the oblique derivative problem, the biharmonic equation, the heat and wave equations and viscoelasticity. ... Read more

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Lucid, self-contained exposition of the theory of ordinary differential equations and integral equations. Especially detailed treatment of the boundary value problem of second order linear ordinary differential equations. Other topics include Fredholm integral equations, Volterra integral equations, much more. Bibliography.
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Integral equations have wide applicability in various fields such as heat and mass transfer, elasticity, plasticity, oscillation theory, fluid dynamics, filtration theory, electrostatics, electrodynamics biomechanics, game theory, control, queuing theory, mechanical and electrical engineering, economics, and medicine. Computational Methods for Linear Integral Equations presents basic theoretical material that deals with numerical analysis, convergence, error estimates and accuracy. The unique computational aspect leads the reader from theoretical and practical problems all the way through to computation with hands-on guidance for input files and the execution of computer programs.This book is an essential reference and authoritative resource for all professionals, graduate students, and researchers in mathematics, physical sciences and engineering. Researchers interested in numerical solution of integral equations will find its practical problem-solving style both accessible and useful for their work. ... Read more

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This monograph investigates algorithms for determining power integral bases in algebraic number fields. The problem has classical roots and leads to the problem of solving the corresponding index form equations that are often reduced to more classical equations, such as various types of Thue equations.

The reader is introduced to the best-known methods for solving several types of diophantine equations using Baker-type estimates, reduction methods, and enumeration algorithms. These methods can be useful for other types of diophantine equations not included in the book.

Several interesting properties of number fields are examined. Some infinite parametric families of fields are also considered as well as the resolution of the corresponding infinite parametric families of diophantine equations. The text is illustrated with several tables of various number fields, including their data on power integral bases.

Advanced undergraduates and graduates will benefit from this exposition of methods for solving some classical types of diophantine equations. Researchers in the field will find new applications for the tools presented throughout the book. ... Read more

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In the present paper a new approach to the numerical solution of Cauchy singular integral equations on the interval by collocation and Galerkin methods is considered. Both methods are based on weighted orthogonal polynomials. The main advantage of our approach, in particular of the collocation method, is the fact that in contrast to usual methods its construction does not depend on the concrete equation and requires less preprocessing. Furthermore, it can also be applied to the system case. On the basis of Banach algebra methods, necessary and sufficient stability conditions are derived, where from the coefficients of the operator only piecewise continuity is required. In a scale of Sobolev spaces we can prove results on convergence rates. Furthermore, in the case of the collocation method we discuss some computational aspects to derive effective algorithms for the fast solution of the approximate equations and present numerical results. In the case of equations perturbed by an integral operator with smooth kernel, we consider a quadrature method and two fast algorithms that allow to make use of the fast solution of the unperturbed equation by the collocation method. ... Read more

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This volume is produced from digital images created through the University of Michigan University Library's large-scale digitization efforts. The Library seeks to preserve the intellectual content of items in a manner that facilitates and promotes a variety of uses. The digital reformatting process results in an electronic version of the original text that can be both accessed online and used to create new print copies. The Library also understands and values the usefulness of print and makes reprints available to the public whenever possible. This book and hundreds of thousands of others can be found in the HathiTrust, an archive of the digitized collections of many great research libraries. For access to the University of Michigan Library's digital collections, please see http://www.lib.umich.edu and for information about the HathiTrust, please visit http://www.hathitrust.org ... Read more

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The effectiveness of dual integral equations for handling mixed boundary value problems has established them as an important tool for applied mathematicians. Their many applications in mathematical physics have prompted extensive research over the last 25 years, and many researchers have made significant contributions to the methodology of solving and to the applications of dual integral equations. However, until now, much of this work has been available only in the form of research papers scattered throughout different journals.In Advances in Dual Integral Equations, the authors systematically present some of the recent developments in dual integral equations involving various special functions as kernel. They examine dual integral equations with Bessel, Legendre, and trigonometric functions as kernel plus dual integral equations involving inverse Mellin transforms. These can be particularly useful in studying certain mixed boundary value problems involving homogeneous media in continuum mechanics. However, when dealing with problems involving non-homogenous media, the corresponding equations may have different kernels. This application prompts the authors to conclude with a discussion of hybrid dual integral equations-mixed kernels with generalized associated Legendre functions and mixed kernels involving Bessel functions.Researchers in the theory of elasticity, fluid dynamics, and mathematical physics will find Advances in Dual Integral Equations a concise, one-stop resource for recent work addressing special functions as kernel. ... Read more

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In many fields of application of mathematics, progress is crucially dependent on the good flow of information between (i) theoretical mathematicians looking for applications, (ii) mathematicians working in applications in need of theory, and (iii) scientists and engineers applying mathematical models and methods. The intention of this book is to stimulate this flow of information. In the first three chapters (accessible to third year students of mathematics and physics and to mathematically interested engineers) applications of Abel integral equations are surveyed broadly including determination of potentials, stereology, seismic travel times, spectroscopy, optical fibres. In subsequent chapters (requiring some background in functional analysis) mapping properties of Abel integral operators and their relation to other integral transforms in various function spaces are investi- gated, questions of existence and uniqueness of solutions of linear and nonlinear Abel integral equations are treated, and for equations of the first kind problems of ill-posedness are discussed. Finally, some numerical methods are described. In the theoretical parts, emphasis is put on the aspects relevant to applications. ... Read more