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An Introduction to Phase Integral Methods

An Introduction to Phase Integral Methods Author John Heading
ISBN-10 9780486497426
Release 2013
Pages 160
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The phase-integral method in mathematics, also known as the Wentzel-Kramers-Brillouin (WKB) method, is the focus of this introductory treatment. Author John Heading successfully steers a course between simplistic and rigorous approaches to provide a concise overview for advanced undergraduates and graduate students in mathematics and physics. Since the number of applications is vast, the text considers only a brief selection of topics and emphasizes the method itself rather than detailed applications. The process, once derived, is shown to be one of essential simplicity that involves merely the application of certain well-defined rules. Starting with a historical survey of the problem and its solutions, subjects include the Stokes phenomenon, one and two transition points, and applications to physical problems. An appendix and bibliography conclude the text.

Introduction to Nonlinear Differential and Integral Equations

Introduction to Nonlinear Differential and Integral Equations Author Harold Thayer Davis
ISBN-10 0486609715
Release 1962
Pages 566
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Topics covered include differential equations of the 1st order, the Riccati equation and existence theorems, 2nd order equations, elliptic integrals and functions, nonlinear mechanics, nonlinear integral equations, more. Includes 137 problems.

Asymptotic Expansions of Integrals

Asymptotic Expansions of Integrals Author Norman Bleistein
ISBN-10 9780486650821
Release 1975
Pages 425
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Excellent introductory text, written by two experts, presents a coherent and systematic view of principles and methods. Topics include integration by parts, Watson's lemma, LaPlace's method, stationary phase, and steepest descents. Additional subjects include the Mellin transform method and less elementary aspects of the method of steepest descents. 1975 edition.

Ray Tracing and Beyond

Ray Tracing and Beyond Author E. R. Tracy
ISBN-10 9780521768061
Release 2014-02-27
Pages 541
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This complete introduction to the use of modern ray tracing techniques in plasma physics describes the powerful mathematical methods generally applicable to vector wave equations in non-uniform media, and clearly demonstrates the application of these methods to simplify and solve important problems in plasma wave theory. Key analytical concepts are carefully introduced as needed, encouraging the development of a visual intuition for the underlying methodology, with more advanced mathematical concepts succinctly explained in the appendices, and supporting Matlab and Raycon code available online. Covering variational principles, covariant formulations, caustics, tunnelling, mode conversion, weak dissipation, wave emission from coherent sources, incoherent wave fields, and collective wave absorption and emission, all within an accessible framework using standard plasma physics notation, this is an invaluable resource for graduate students and researchers in plasma physics.

Techniques and Applications of Path Integration

Techniques and Applications of Path Integration Author L. S. Schulman
ISBN-10 9780486137025
Release 2012-10-10
Pages 448
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Suitable for advanced undergraduates and graduate students, this text develops the techniques of path integration and deals with applications, covering a host of illustrative examples. 26 figures. 1981 edition.

Differential and Integral Equations

Differential and Integral Equations Author Peter J. Collins
ISBN-10 9780198533825
Release 2006-08-03
Pages 372
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Differential & integral equations involve important mathematical techniques, & as such will be encountered by mathematicians, & physical & social scientists, in their undergraduate courses. This text provides a clear, comprehensive guide to first- & second- order ordinary & partial differential equations.

Advanced Mathematical Methods for Scientists and Engineers I

Advanced Mathematical Methods for Scientists and Engineers I Author Carl M. Bender
ISBN-10 0387989315
Release 1999-10-29
Pages 593
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This book gives a clear, practical and self-contained presentation of the methods of asymptotics and perturbation theory for obtaining approximate analytical solutions to differential and difference equations. These methods allow one to analyze physics and engineering problems that may not be solvable in closed form. The presentation provides insights that will be useful in approaching new problems.

Applied Complex Variables

Applied Complex Variables Author John W. Dettman
ISBN-10 9780486158280
Release 2012-05-07
Pages 512
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Fundamentals of analytic function theory — plus lucid exposition of 5 important applications: potential theory, ordinary differential equations, Fourier transforms, Laplace transforms, and asymptotic expansions. Includes 66 figures.

A Modern Approach to Functional Integration

A Modern Approach to Functional Integration Author John R. Klauder
ISBN-10 9780817647902
Release 2010-11-17
Pages 282
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This text takes advantage of recent developments in the theory of path integration and attempts to make a major paradigm shift in how the art of functional integration is practiced. The techniques developed in the work will prove valuable to graduate students and researchers in physics, chemistry, mathematical physics, and applied mathematics who find it necessary to deal with solutions to wave equations, both quantum and beyond. A Modern Approach to Functional Integration offers insight into a number of contemporary research topics, which may lead to improved methods and results that cannot be found elsewhere in the textbook literature. Exercises are included in most chapters, making the book suitable for a one-semester graduate course on functional integration.

A Combinatorial Introduction to Topology

A Combinatorial Introduction to Topology Author Michael Henle
ISBN-10 0486679667
Release 1979
Pages 310
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Excellent text covers vector fields, plane homology and the Jordan Curve Theorem, surfaces, homology of complexes, more. Problems and exercises. Some knowledge of differential equations and multivariate calculus required.Bibliography. 1979 edition.

Asymptotic Analysis of Differential Equations

Asymptotic Analysis of Differential Equations Author R. B. White
ISBN-10 9781848166080
Release 2010
Pages 405
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"This is a useful volume in which a wide selection of asymptotic techniques is clearly presented in a form suitable for both applied mathematicians and Physicists who require an introduction to asymptotic techniques." --Book Jacket.

An Introduction to Differential Equations and Their Applications

An Introduction to Differential Equations and Their Applications Author Stanley J. Farlow
ISBN-10 9780486135137
Release 2012-10-23
Pages 640
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This introductory text explores 1st- and 2nd-order differential equations, series solutions, the Laplace transform, difference equations, much more. Numerous figures, problems with solutions, notes. 1994 edition. Includes 268 figures and 23 tables.

Modelling with Differential and Difference Equations

Modelling with Differential and Difference Equations Author Glenn Fulford
ISBN-10 052144618X
Release 1997-06-12
Pages 405
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The theme of this book is modeling the real world using mathematics. The authors concentrate on the techniques used to set up mathematical models and describe many systems in full detail, covering both differential and difference equations in depth. Among the broad spectrum of topics studied in this book are: mechanics, genetics, thermal physics, economics and population studies.

Path Integrals in Quantum Mechanics Statistics Polymer Physics and Financial Markets

Path Integrals in Quantum Mechanics  Statistics  Polymer Physics  and Financial Markets Author Hagen Kleinert
ISBN-10 9789814273572
Release 2009
Pages 1624
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This is the fifth, expanded edition of the comprehensive textbook published in 1990 on the theory and applications of path integrals. It is the first book to explicitly solve path integrals of a wide variety of nontrivial quantum-mechanical systems, in particular the hydrogen atom. The solutions have been made possible by two major advances. The first is a new euclidean path integral formula which increases the restricted range of applicability of Feynman''s time-sliced formula to include singular attractive 1/r- and 1/r2-potentials. The second is a new nonholonomic mapping principle carrying physical laws in flat spacetime to spacetimes with curvature and torsion, which leads to time-sliced path integrals that are manifestly invariant under coordinate transformations. In addition to the time-sliced definition, the author gives a perturbative, coordinate-independent definition of path integrals, which makes them invariant under coordinate transformations. A consistent implementation of this property leads to an extension of the theory of generalized functions by defining uniquely products of distributions. The powerful Feynman-Kleinert variational approach is explained and developed systematically into a variational perturbation theory which, in contrast to ordinary perturbation theory, produces convergent results. The convergence is uniform from weak to strong couplings, opening a way to precise evaluations of analytically unsolvable path integrals in the strong-coupling regime where they describe critical phenomena. Tunneling processes are treated in detail, with applications to the lifetimes of supercurrents, the stability of metastable thermodynamic phases, and the large-order behavior of perturbation expansions. A variational treatment extends the range of validity to small barriers. A corresponding extension of the large-order perturbation theory now also applies to small orders. Special attention is devoted to path integrals with topological restrictions needed to understand the statistical properties of elementary particles and the entanglement phenomena in polymer physics and biophysics. The ChernoOe1/4OC Simons theory of particles with fractional statistics (anyons) is introduced and applied to explain the fractional quantum Hall effect. The relevance of path integrals to financial markets is discussed, and improvements of the famous BlackoOe1/4OC Scholes formula for option prices are developed which account for the fact, recently experienced in the world markets, that large fluctuations occur much more frequently than in Gaussian distributions."

Geometric Numerical Integration

Geometric Numerical Integration Author Ernst Hairer
ISBN-10 9783662050187
Release 2013-03-09
Pages 515
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This book deals with numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions. 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 numerous figures, treats applications from physics and astronomy, and contains many numerical experiments and comparisons of different approaches.

The Qualitative Theory of Ordinary Differential Equations

The Qualitative Theory of Ordinary Differential Equations Author Fred Brauer
ISBN-10 9780486151519
Release 2012-12-11
Pages 320
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Superb, self-contained graduate-level text covers standard theorems concerning linear systems, existence and uniqueness of solutions, and dependence on parameters. Focuses on stability theory and its applications to oscillation phenomena, self-excited oscillations, more. Includes exercises.

Linear Differential Operators

Linear Differential Operators Author Cornelius Lanczos
ISBN-10 0486680355
Release 1997
Pages 564
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The basic and characteristic properties of linear differential operators are explored in this graduate-level text. No specific knowledge beyond the usual introductory courses is necessary. Includes 350 problems and solution.