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bender c advanced mathematical methods scientists engineers i 1999

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Textbook in PDF format "In this volume we are concerned only with functions of one variable. We hope some day to write a sequel to this book on partial differential equations." — But unfortunately Part II on Partial Differential Equations never appeared! This book is a reprint of the original published by McGraw-Hill in 1978. The changes are the addition of the Roman numeral I to the title and the provision of a subtitle, "Asymptotic Methods and Perturbation Theory". The contents are the same, but some problems are different. Provision of a subtitle may be needed. It is an excellent introduction to asymptotic and perturbation methods for master's degree students or beginning research students. Certain parts of it could be used for a course in asymptotics for final year undergraduates in applied mathematics or mathematical physics. This is a book that has stood the test of time. It is written in a fresh and lively style and the many graphs and tables, comparing the results of exact and approximate methods, were in advance of its time. Springer-Verlag has done a great service to users of, and researchers in, asymptotics and perturbation theory by reprinting this classic. 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. Aimed at teaching the most useful insights in approaching new problems. Intended for graduates and advanced undergraduates, it assumes only a limited familiarity with differential equations and complex variables. The presentation begins with a review of differential and difference equations, then develops local asymptotic methods for such equations, and explains perturbation and summation theory before concluding with an exposition of global asymptotic methods. Emphasizing applications, the discussion stresses care rather than rigor and relies on many well-chosen examples to teach readers how an applied mathematician tackles problems. There are 190 computer-generated plots and tables comparing approximate and exact solutions, over 600 problems of varying levels of difficulty, and an appendix summarizing the properties of special functions. The main purpose of our book is to present and explain mathematical methods for obtaining approximate analytical solutions to differential and difference equations that cannot be solved exactly. Our objective is to help young and also established scientists and engineers to build the skills necessary to analyze equations that they encounter in their work. Our presentation is aimed at developing the insights and techniques that are most useful for attacking new problems. We do not emphasize special methods and tricks which work only for the classical transcendental functions; we do not dwell on equations whose exact solutions are known. The mathematical methods discussed in this book are known collectively as asymptotic and perturbative analysis. These are the most useful and powerful methods for finding approximate solutions to equations, but they are difficult to justify rigorously. Thus, we concentrate on the most fruitful aspect of applied analysis; namely, obtaining the answer. We stress care but not rigor. Asymptotics is a newer calculus, an approximate calculus, and its mathematical subtleties are as difficult for an advanced student as the subtleties of calculus are for a freshman. This volume teaches the new kind of approximate calculus necessary to solve hard problems approximately. We believe that our book is the first comprehensive book at the advanced undergraduate or beginning graduate level that has this kind of problem-solving approach to applied mathematics. We recognize that the readership of this book will be extremely diverse. Therefore, we have organized the book so that it will be useful to beginning students as well as to experienced researchers. First, this book is completely self-contained. We have included a review of ordinary differential equations and ordinary difference equations in Part I for those readers whose background is weak. There is also an Appendix of useful formulas so that it will rarely be necessary to consult outside reference books on special functions. Second, we indicate the difficulty of every section by the three letters E (easy), I (intermediate), and D (difficult). We also use the letter T to indicate that the material has a theoretical as opposed to an applied or calculational slant. We have rated the material this way to help readers and teachers to select the level of material that is appropriate for their needs. We have included a large selection of exercises and problems at the end of each chapter. The difficulty and slant of each problem is also indicated by the letters E, I, D, and T. A good undergraduate course on mathematical methods can be based entirely on the sections and problems labeled E. One of the novelties of this book is that we illustrate the results of our asymptotic analysis graphically by presenting many computer plots and tables which compare exact and approximate answers. These plots and tables should give the reader a feeling of just how well approximate analytical methods work. It is our experience that these graphs are an effective teaching device that strengthens the reader's belief that approximation methods can be usefully applied to the problems that he or she need to solve. Fundamentals Ordinary Differential Equations Difference Equations Local Analysis Approximate Solution of Linear Differential Equations Approximate Solution of Nonlinear Differential Equations Approximate Solution of Difference Equations Asymptotic Expansion of Integrals Perturbation Methods Perturbation Series Summation of Series Global Analysis Boundary Layer Theory WKB Theory Multiple-Scale Analysis Appendix - Useful Formulas References Index

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