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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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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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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 methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are the subject of this book. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by many figures, it treats applications from physics and astronomy and contains many numerical experiments and comparisons of different approaches. The second edition is substantially revised and enlarged, with many improvements in the presentation and additions concerning in particular non-canonical Hamiltonian systems, highly oscillatory mechanical systems, and the dynamics of multistep methods.
Numerical integration. --- Hamiltonian systems. --- Differential equations --- Numerical solutions. --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Numerical analysis. --- Biomathematics. --- Physics. --- Numerical Analysis. --- Analysis. --- Theoretical, Mathematical and Computational Physics. --- Mathematical Methods in Physics. --- Numerical and Computational Physics. --- Mathematical and Computational Biology. --- 517.91 Differential equations --- Hamiltonian dynamical systems --- Systems, Hamiltonian --- Differentiable dynamical systems --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Definite integrals --- Interpolation --- Numerical analysis --- Global analysis (Mathematics). --- Mathematical physics. --- Numerical and Computational Physics, Simulation. --- Physical mathematics --- Physics --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematical analysis --- Mathematics --- Biology --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- 517.1 Mathematical analysis --- Hamiltonian systems --- Differential equations - Numerical solutions --- 517.91 --- Numerical integration --- 519.62 --- 681.3*G17 --- 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) --- 519.62 Numerical methods for solution of ordinary differential equations --- Numerical methods for solution of ordinary differential equations --- Numerical solutions
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Asymptotic Wave Theory
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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