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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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Mathematical analysis --- Riemann integral --- Intégrale de Riemann --- 517.1 --- Integral, Riemann --- Definite integrals --- Introduction to analysis --- Riemann integral. --- 517.1 Introduction to analysis --- Intégrale de Riemann
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Finite element method. --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Differential equations --- Numerical integration. --- Numerical solutions. --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Definite integrals --- Interpolation --- Numerical analysis --- 517.91 Differential equations
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Ce Petit traité d’intégration développe une approche originale de l’intégrale. Cette approche, que l’on pourrait qualifier de globale, est due aux deux mathématiciens Jaroslaw Kurzweil et Ralph Henstock. L’enseignement de l’intégration se fait d’ordinaire en deux temps. On débute en proposant des approximations de l’aire située sous le graphe de la fonction sous la forme de sommes de Riemann, ce qui est bien adapté au calcul différentiel et intégral portant sur des fonctions régulières. On présente ensuite l’intégrale de Lebesgue en lien avec la théorie de la mesure. L’approche de Kurzweil et Henstock est proche de celle de Riemann, à cela près que le pas des subdivisions de l’intervalle pour le calcul de l’aire peut ne pas être constant. L’intérêt de cette méthode est de contenir la théorie de Lebesgue et d’être optimale pour le calcul différentiel. Ce livre concerne au premier chef les étudiants de mathématiques de tous les cycles (licence, master, préparation aux concours de l’enseignement…). Il intéressera également les enseignants de mathématiques ou de physique et, plus généralement, les ingénieurs et scientifiques qui font usage de la théorie de l’intégration.
Integration, Functional. --- Riemann integral. --- Henstock-Kurzweil integral. --- Riemann, Bernhard, --- Lebesgue, Henri Léon, --- Gauge integral --- Generalized Riemann integral --- Henstock integrals --- HK integral --- Kurzweil-Henstock integral --- Kurzweil integral --- Riemann integral, Generalized --- Integral, Riemann --- Functional integration --- Lebeg, Anri, --- Riemann, B. --- Riman, Georg Fridrikh Bernkhard, --- Riman, Bernkhard, --- Riemann, Georg Friedrich Bernhard, --- Integrals, Generalized --- Definite integrals --- Functional analysis --- Lebesgue, Henri, --- Functions of several complex variables. --- Fonctions de plusieurs variables complexes. --- Intégration de fonctions.
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681.3*G14 --- Quadrature and numerical differentiation: adaptive quadrature; equal intervalintegration; error analysis; finite difference methods; gaussian quadrature; iterated methods; multiple quadrature --- 681.3*G14 Quadrature and numerical differentiation: adaptive quadrature; equal intervalintegration; error analysis; finite difference methods; gaussian quadrature; iterated methods; multiple quadrature --- Numerical integration --- 519.6 --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- Computational mathematics. Numerical analysis. Computer programming --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Definite integrals --- Interpolation --- Numerical analysis
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Indispensable for students, invaluable for researchers, this comprehensive treatment of contemporary quasi-Monte Carlo methods, digital nets and sequences, and discrepancy theory starts from scratch with detailed explanations of the basic concepts and then advances to current methods used in research. As deterministic versions of the Monte Carlo method, quasi-Monte Carlo rules have increased in popularity, with many fruitful applications in mathematical practice. These rules require nodes with good uniform distribution properties, and digital nets and sequences in the sense of Niederreiter are known to be excellent candidates. Besides the classical theory, the book contains chapters on reproducing kernel Hilbert spaces and weighted integration, duality theory for digital nets, polynomial lattice rules, the newest constructions by Niederreiter and Xing and many more. The authors present an accessible introduction to the subject based mainly on material taught in undergraduate courses with numerous examples, exercises and illustrations.
Monte Carlo method. --- Nets (Mathematics) --- Sequences (Mathematics) --- Numerical integration. --- Digital filters (Mathematics) --- Data smoothing filters --- Filters, Digital (Mathematics) --- Linear digital filters (Mathematics) --- Linear filters (Mathematics) --- Numerical filters --- Smoothing filters (Mathematics) --- Digital electronics --- Filters (Mathematics) --- Fourier transformations --- Functional analysis --- Numerical analysis --- Numerical calculations --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Definite integrals --- Interpolation --- Mathematical sequences --- Numerical sequences --- Algebra --- Mathematics --- Moore-Smith convergence --- Net equations --- Net methods (Mathematics) --- Convergence --- Set theory --- Topology --- Artificial sampling --- Model sampling --- Monte Carlo simulation --- Monte Carlo simulation method --- Stochastic sampling --- Games of chance (Mathematics) --- Mathematical models --- Stochastic processes
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Numerical solutions of differential equations --- Differential equations --- Numerical solutions --- -Numerical integration --- 519.6 --- 681.3*G14 --- 681.3*G17 --- Equations, Differential --- Bessel functions --- Calculus --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Definite integrals --- Interpolation --- Numerical analysis --- Computational mathematics. Numerical analysis. Computer programming --- Quadrature and numerical differentiation: adaptive quadrature; equal intervalintegration; error analysis; finite difference methods; gaussian quadrature; iterated methods; multiple quadrature --- Ordinary differential equations: boundary value problems; convergence and stability; error analysis; initial value problems; multistep methods; single step methods; stiff equations (Numerical analysis) --- Numerical integration. --- Numerical solutions. --- 517.91 Differential equations --- 681.3*G17 Ordinary differential equations: boundary value problems; convergence and stability; error analysis; initial value problems; multistep methods; single step methods; stiff equations (Numerical analysis) --- 681.3*G14 Quadrature and numerical differentiation: adaptive quadrature; equal intervalintegration; error analysis; finite difference methods; gaussian quadrature; iterated methods; multiple quadrature --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- Numerical integration --- 517.91 --- Initial value problems --- Équations différentielles --- Problèmes aux valeurs initiales --- Numerical solutions&delete& --- Analyse numérique. --- Équations différentielles. --- Problèmes aux valeurs initiales. --- Differential equations - Numerical solutions
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Construction Of Integration Formulas For Initial Value Problems
Mathematical physics --- Electromagnetism. Ferromagnetism --- Differential equations --- Initial value problems --- Équations différentielles. --- Problèmes aux valeurs initiales. --- Développements asymptotiques --- Col, Méthode du --- Asymptotic expansions --- Method of steepest descent (Numerical analysis) --- Numerical integration --- Problèmes aux valeurs initiales --- Intégration numérique --- Wave-motion, Theory of. --- Wave equation. --- Asymptotic expansions. --- Initial value problems. --- Numerical integration. --- Wave equations --- Asymptotic theory. --- Asymptotic developments --- Asymptotes --- Convergence --- Difference equations --- Divergent series --- Functions --- Numerical analysis --- Differential equations, Partial --- Wave-motion, Theory of --- Undulatory theory --- Mechanics --- 519.6 --- 681.3*G17 --- 681.3 *G18 --- 532 --- 532 Fluid mechanics in general. Mechanics of liquids (hydromechanics) --- Fluid mechanics in general. Mechanics of liquids (hydromechanics) --- Développements asymptotiques --- Col, Méthode du --- Ondes --- Propagation --- Differential equations. --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- Computational mathematics. Numerical analysis. Computer programming --- 681.3 *G18 Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- 681.3*G17 Ordinary differential equations: boundary value problems; convergence and stability; error analysis; initial value problems; multistep methods; single step methods; stiff equations (Numerical analysis) --- Ordinary differential equations: boundary value problems; convergence and stability; error analysis; initial value problems; multistep methods; single step methods; stiff equations (Numerical analysis) --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Definite integrals --- Interpolation --- Problems, Initial value --- Boundary value problems --- Analyse numerique --- Equations differentielles ordinaires --- Equations differentielles --- Methodes numeriques --- Equations aux derivees partielles hyperboliques --- Transformation de laplace --- Geophysique --- Geodynamique --- Seismologie
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Chromosomes --- Karyotypes --- Congresses --- 681.5.016 --- -Computer algorithms --- Differential equations --- -Iterative methods (Mathematics) --- Numerical integration --- Stiff computation (Differential equations) --- #TELE:d.d. Prof. A. J. J. Oosterlinck --- Computation, Stiff (Differential equations) --- Equations, Stiff (Differential equations) --- Stiff equations (Differential equations) --- Stiff systems (Differential equations) --- Systems, Stiff (Differential equations) --- Differential equations, Partial --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Definite integrals --- Interpolation --- Numerical analysis --- Iteration (Mathematics) --- Equations, Differential --- Bessel functions --- Calculus --- Algorithms --- Chromosome theory --- Cell nuclei --- Crossing over (Genetics) --- Cytotaxonomy --- Genetics --- Karyokinesis --- Linkage (Genetics) --- Automatic control engineering. Control systems, techniques, equipment. Cybernetic and automation technology--?.016 --- Numerical solutions --- Computer algorithms. --- Iterative methods (Mathematics) --- Numerical integration. --- Numerical solutions. --- 517.91 Differential equations --- 681.5.016 Automatic control engineering. Control systems, techniques, equipment. Cybernetic and automation technology--?.016 --- Iterative methods (Mathematics). --- Stiff computation (Differential equations). --- -Chromosomes --- Ordinary differential equations: boundary value problems; convergence and stability; error analysis; initial value problems; multistep methods; single step methods; stiff equations (Numerical analysis) --- 681.3*G17 Ordinary differential equations: boundary value problems; convergence and stability; error analysis; initial value problems; multistep methods; single step methods; stiff equations (Numerical analysis) --- -519.6 --- 681.3*G17 --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- Computational mathematics. Numerical analysis. Computer programming --- Chromosome abnormalities --- Numerical solutions of differential equations --- 517.91 --- Initial value problems --- Équations différentielles --- Problèmes aux valeurs initiales --- Numerical solutions&delete& --- Équations différentielles. --- Problèmes aux valeurs initiales. --- Differential equations. --- Initial value problems. --- Problèmes aux valeurs initiales --- Karyotypes - Congresses --- Chromosomes - Congresses --- Analyse numerique --- Equations differentielles --- -Congresses --- Chromosomes, human
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