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Asymptotic Approximations of Integrals deals with the methods used in the asymptotic approximation of integrals. Topics covered range from logarithmic singularities and the summability method to the distributional approach and the Mellin transform technique for multiple integrals. Uniform asymptotic expansions via a rational transformation are also discussed, along with double integrals with a curve of stationary points. For completeness, classical methods are examined as well. Comprised of nine chapters, this volume begins with an introduction to the fundamental concepts of asymptotics, followed by a discussion on classical techniques used in the asymptotic evaluation of integrals, including Laplace's method, Mellin transform techniques, and the summability method. Subsequent chapters focus on the elementary theory of distributions; the distributional approach; uniform asymptotic expansions; and integrals which depend on auxiliary parameters in addition to the asymptotic variable. The book concludes by considering double integrals and higher-dimensional integrals. This monograph is intended for graduate students and research workers in mathematics, physics, and engineering.
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An attractive book on the intersection of analysis and numerical analysis, deriving classical boundary integral equations arising from the potential theory and acoustics. This self-contained monograph can be used as a textbook by graduate/postgraduate students. It also contains a lot of carefully chosen exercises.
Methods of Numerical Approximation is based on lectures delivered at the Summer School held in September 1965, at Oxford University. The book deals with the approximation of functions with one or more variables, through means of more elementary functions. It explains systems to approximate functions, such as trigonometric sums, rational functions, continued fractions, and spline functions. The book also discusses linear approximation including topics such as convergence of polynomial interpolation and the least-squares approximation. The text analyzes Bernstein polynomials, Weierstrass' theorem, and Lagrangian interpolation. The book also gives attention to the Chebyshev least-squares approximation, the Chebyshev series, and the determination of Chebyshev series, under general methods. These general methods are useful when the student wants to investigate practical methods for finding forms of approximations under various situations. One of the lectures concerns the general theory of linear approximation and the existence of a best approximation approach using different theorems. The book also discusses the theory and calculation of the best rational approximations as well as the optimal approximation of linear functionals. The text will prove helpful for students in advanced mathematics and calculus. It can be appreciated by statisticians and those working with numbers theory.
Integration in infinitely dimensional spaces (continual integration) is a powerful mathematical tool which is widely used in a number of fields of modern mathematics, such as analysis, the theory of differential and integral equations, probability theory and the theory of random processes. This monograph is devoted to numerical approximation methods of continual integration. A systematic description is given of the approximate computation methods of functional integrals on a wide class of measures, including measures generated by homogeneous random processes with independent increments and Gaussian processes. Many applications to problems which originate from analysis, probability and quantum physics are presented. This book will be of interest to mathematicians and physicists, including specialists in computational mathematics, functional and statistical physics, nuclear physics and quantum optics.
Methods of Numerical Integration, Second Edition describes the theoretical and practical aspects of major methods of numerical integration. Numerical integration is the study of how the numerical value of an integral can be found. This book contains six chapters and begins with a discussion of the basic principles and limitations of numerical integration. The succeeding chapters present the approximate integration rules and formulas over finite and infinite intervals. These topics are followed by a review of error analysis and estimation, as well as the application of functional analysis to numerical integration. A chapter describes the approximate integration in two or more dimensions. The final chapter looks into the goals and processes of automatic integration, with particular attention to the application of Tschebyscheff polynomials. This book will be of great value to theoreticians and computer programmers.
This is the first book devoted to lattice methods, a recently developed way of calculating multiple integrals in many variables. Multiple integrals of this kind arise in fields such as quantum physics and chemistry, statistical mechanics, Bayesian statistics and many others. Lattice methods are an effective tool when the number of integrals are large. The book begins with a review of existing methods before presenting lattice theory in a thorough, self-contained manner, with numerous illustrations and examples. Group and number theory are included, but the treatment is such that no prior knowledge is needed. Not only the theory but the practical implementation of lattice methods is covered. An algorithm is presented alongside tables not available elsewhere, which together allow the practical evaluation of multiple integrals in many variables. Most importantly, the algorithm produces an error estimate in a very efficient manner. The book also provides a fast track for readers wanting to move rapidly to using lattice methods in practical calculations. It concludes with extensive numerical tests which compare lattice methods with other methods, such as the Monte Carlo.