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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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Riemann integral. --- Integral, Riemann --- Definite integrals --- Integral de Riemann --- Integrals definides

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This textbook provides a thorough introduction to measure and integration theory, fundamental topics of advanced mathematical analysis. Proceeding at a leisurely, student-friendly pace, the authors begin by recalling elementary notions of real analysis before proceeding to measure theory and Lebesgue integration. Further chapters cover Fourier series, differentiation, modes of convergence, and product measures. Noteworthy topics discussed in the text include Lp spaces, the Radon–Nikodým Theorem, signed measures, the Riesz Representation Theorem, and the Tonelli and Fubini Theorems. This textbook, based on extensive teaching experience, is written for senior undergraduate and beginning graduate students in mathematics. With each topic carefully motivated and hints to more than 300 exercises, it is the ideal companion for self-study or use alongside lecture courses.

Mathematics. --- Fourier analysis. --- Functional analysis. --- Measure and Integration. --- Real Functions. --- Fourier Analysis. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Analysis, Fourier --- Mathematical analysis --- Math --- Science --- Riemann integral. --- Integral, Riemann --- Definite integrals --- Measure theory. --- Functions of real variables. --- Real variables --- Functions of complex variables --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra)

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This book deals with the numerical analysis and efficient numerical treatment of high-dimensional integrals using sparse grids and other dimension-wise integration techniques with applications to finance and insurance. The book focuses on providing insights into the interplay between coordinate transformations, effective dimensions and the convergence behaviour of sparse grid methods. The techniques, derivations and algorithms are illustrated by many examples, figures and code segments. Numerical experiments with applications from finance and insurance show that the approaches presented in this book can be faster and more accurate than (quasi-) Monte Carlo methods, even for integrands with hundreds of dimensions.

Calculus, Integral. --- Numerical grid generation (Numerical analysis). --- Numerical analysis --- Finance --- Risk --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Applied Mathematics --- Mathematics - General --- Mathematical models --- Numerical integration. --- Insurance --- Mathematical models. --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Mathematics. --- Economics, Mathematical. --- Computer mathematics. --- Computational Mathematics and Numerical Analysis. --- Quantitative Finance. --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Economics --- Mathematical economics --- Econometrics --- Math --- Science --- Methodology --- Definite integrals --- Interpolation --- Computer science --- Finance. --- Funding --- Funds --- Currency question --- Economics, Mathematical .

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Numerical integration --- 519.6 --- 681.3*G14 --- 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 --- Numerical integration. --- 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 --- Analyse numérique. --- Analyse numérique --- Numerical analysis. --- Integration numerique --- Calcul integral --- Integrales multiples