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The derivative and the integral are the fundamental notions of calculus. Though there is essentially only one derivative, there are 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. Historical information is plentiful. Advanced undergraduate mathematics majors, graduate students, and faculty members are the audience for the book. 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 the richness of the idea of integral. Professor Burke's clear and well-motivated exposition makes this book a joy to read.
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The problem of evaluating integrals is well known to every student who has had a year of calculus. It was an especially important subject in 19th century analysis and it has now been revived with the appearance of symbolic languages. In this book, the authors use the problem of exact evaluation of definite integrals as a starting point for exploring many areas of mathematics. The questions discussed in this book, first published in 2004, are as old as calculus itself. In presenting the combination of methods required for the evaluation of most integrals, the authors take the most interesting, rather than the shortest, path to the results. Along the way, they illuminate connections with many subjects, including analysis, number theory, algebra and combinatorics. This will be a guided tour of exciting discovery for undergraduates and their teachers in mathematics, computer science, physics, and engineering.
Definite integrals. --- Integrals. --- Calculus, Integral --- Integrals, Definite --- Integrals
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Calculus. --- Calculus, Integral. --- Integrals. --- Calculus, Integral --- Integral calculus --- Differential equations --- Analysis (Mathematics) --- Fluxions (Mathematics) --- Infinitesimal calculus --- Limits (Mathematics) --- Mathematical analysis --- Functions --- Geometry, Infinitesimal
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This work deals with integrals in calculus which is a fundamental tool for mathematical analysis and for the calculation of probabilities. The work is divided into 11 chapters as follows: Computations using integrals, for example Lebesgue integration techniques as it applies to measurable real analysis, measurable spaces, for example Euclidean n-space, integrable functions defined on Rn, the calculations of Lebesgue-Stieltjes integration, their functions as defined by integrals, convolution, Fourier transform, Fourier series, the applications and their complements.
Calculus, Integral. --- Differential equations. --- 517.91 Differential equations --- Differential equations --- Integral calculus
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The relatively new concepts of the Henstock-Kurzweil and McShane integrals based on Riemann type sums are an interesting challenge in the study of integration of Banach space-valued functions. This timely book presents an overview of the concepts developed and results achieved during the past 15 years. The Henstock-Kurzweil and McShane integrals play the central role in the book. Various forms of the integration are introduced and compared from the viewpoint of their generality. Functional analysis is the main tool for presenting the theory of summation gauge integrals.
Banach spaces. --- Integrals. --- Calculus, Integral --- Functions of complex variables --- Generalized spaces --- Topology
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Difference equations are playing an increasingly important role in the natural sciences. Indeed many phenomena are inherently discrete and are naturally described by difference equations. Phenomena described by differential equations are therefore approximations of more basic discrete ones. Moreover, in their study it is very often necessary to resort to numerical methods. This always involves a discretization of the differential equations involved, thus replacing them by difference equations. This book shows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference ones. Each of the eleven chapters is a self-contained treatment of a topic, containing introductory material as well as the latest research results. The book will be welcomed by graduate students and researchers seeking an introduction to the field. As a survey of the current state of the art it will also serve as a valuable reference.
Difference equations. --- Symmetry (Mathematics) --- Integrals. --- Calculus, Integral --- Invariance (Mathematics) --- Group theory --- Automorphisms --- Calculus of differences --- Differences, Calculus of --- Equations, Difference
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Intégration, Chapitres 1 à 4 Les Éléments de mathématique de Nicolas BOURBAKI ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce sixième chaptire du Livre d’Intégration, sixième Livre des éléments de mathématique, étend la notion d’intégration à des mesure à valeurs dans des espaces vectoriels de Hausdorff localement convexes. Il contient également une note historique. Ce volume est une réimpression de l’édition de 1959.
Integrals. --- Calculus, Integral --- Mathematics. --- Measure and Integration. --- Math --- Science --- Measure theory. --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra)
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Intégration, Chapitre 5 Les Éléments de mathématique de Nicolas BOURBAKI ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce cinquième chaptire du Livre d’Intégration, sixième Livre des éléments de mathématique, traite notamment d’une generalisation du théorème des Lebesgue-Fubini et du théorème de Lebesque-Nikodym. Il contient également des notes historiques. Ce volume est une réimpression de l’édition de 1967.
Integrals. --- Calculus, Integral --- Mathematics. --- Measure and Integration. --- Math --- Science --- Measure theory. --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra)
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Intégration, Chapitres 1 à 4 Les Éléments de mathématique de Nicolas BOURBAKI ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce premier volume du Livre d’Intégration, sixième Livre du traité, est consacré aux fondements de la théorie de l’intégration, il comprend les chapitres : Inégalités de convexité ; Espaces de Riesz ; Mesures sur les espaces localement compacts ; Prolongement d’une mesure. Espaces Lp. Il contient également une note historique. Ce volume est une réimpression de l’édition de 1965.
Integrals. --- Calculus, Integral --- Mathematics. --- Measure and Integration. --- Math --- Science --- Measure theory. --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra)
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This book gives a straightforward introduction to the field as it is nowadays required in many branches of analysis and especially in probability theory. The first three chapters (Measure Theory, Integration Theory, Product Measures) basically follow the clear and approved exposition given in the author's earlier book on ""Probability Theory and Measure Theory"". Special emphasis is laid on a complete discussion of the transformation of measures and integration with respect to the product measure, convergence theorems, parameter depending integrals, as well as the Radon-Nikodym theorem.
Measure theory. --- Integrals, Generalized. --- Calculus, Integral --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra)
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