Applied Analysis (Dover Books on Mathematics) by Cornelius Lanczos (PDF)

Description

This is a basic text for graduate and advanced undergraduate study in those areas of mathematical analysis that are of primary concern to the engineer and the physicist, most particularly analysis and design of finite processes that approximate the solution of an analytical problem. The work comprises seven chapters:Chapter I (Algebraic Equations) deals with the search for roots of algebraic equations encountered in vibration and flutter problems and in those of static and dynamic stability. Useful computing techniques are discussed, in particular the Bernoulli method and its ramifications.Chapter II (Matrices and Eigenvalue Problems) is devoted to a systematic development of the properties of matrices, especially in the context of industrial research.Chapter III (Large-Scale Linear Systems) discusses the “spectroscopic method” of finding the real eigenvalues of large matrices and the corresponding method of solving large-scale linear equations as well as an additional treatment of a perturbation problem and other topics.Chapter IV (Harmonic Analysis) deals primarily with the interpolation aspects of the Fourier series and its flexibility in representing empirically given equidistant data.Chapter V (Data Analysis) deals with the problem of reduction of data and of obtaining the first and even second derivatives of an empirically given function — constantly encountered in tracking problems in curve-fitting problems. Two methods of smoothing are discussed: smoothing in the small and smoothing in the large.Chapter VI (Quadrature Methods) surveys a variety of quadrature methods with particular emphasis on Gaussian quadrature and its use in solving boundary value problems and eignenvalue problems associated with ordinary differential equations.Chapter VII (Power Expansions) discusses the theory of orthogonal function systems, in particular the “Chebyshev polynomials.”This unique work, perennially in demand, belongs in the library of every engineer, physicist, or scientist interested in the application of mathematical analysis to engineering, physical, and other practical problems.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Applied Analysis, by Cornelius Lanczos is, in the author’s words in the Preface, that branch of analysis devoted to the analysis of finite algorithms, or “workable mathematics”. Today it would be called numerical analysis.Written in 1956,foloowing assignments with then North american Aviation and the Boeing Airplane Company, the book is a compendium of compuational techniques for the solution of cubic, quartic and higher order algebraic equations, matrices and eigenvalue problems, large scale linear systems, harmonic analyis. An entire chapter of 65 pages is devoted to data analysis, the problems associates with processing large amounts of data, and some of the hidden dangers of straightforward(equidistant) interpolation. A chapter is devoted to quadrature methods, and the book concludes with a chapter on power expansions. Most of the computational methods are devoted to hand calculation, or electro mechanical calculation. Today, algorithms developed specifically for high speed digital processors make most of these methods obsolete.I bought the book because of an interest in Legendre polynomials, and their use in fitting to data. However, fascinating tidbits (at least to me) pop up unexpectedly. One tidbit that caught my eye was in the chapter on Data Analysis(Chapter V) in which the Sturm-Liouville equation suddenly appears, is solved by an application of Green’s identity, and hence the result is a proof of the orthoganility of a class of equations (Legendre’s being just one of several) in just two steps. Perhaps I missed something when I studied Series and Special Functions, but I’ve not seen this anywhere else.A very pleasant characteristic of the book is the almost seamless movement back and forth between theory and computationl application. On the other hand, be prepared to spend time with a pencil and paper following not only the derivations and proofs, but the computational algorithms. For someone doing, or planning to do, a massive amount of number crunching of large collections of physical date, this is probably a good book to have.

⭐The book arrived earlier than I expected, and it was easy for me to get it. The book is great and I like it.

⭐It’s an excellent book. The best parts forwe were the chapters on Matrices and onHarmonic Analysis. An outstanding aspectof the latter chapter is Lanczos’s expositionof the motivation behind the Fourier integral(transform) and its basic theory. The qualityof the writing is superb, very classicaland lucid.It cannot, of course, serve as a textbook.But if you’re taking a Fourier theorycourse using Stein and Shakarchi’s book, say,as I am currently, then it’s a very handybook that can complement abstract theorywith physical intuition.

⭐Lanczos’ work is a fine, thorough text that covers most areas of advanced analysis in a readable style. His derivations are clear, his tangential explorations are absorbing, and he cites practical examples. The one area in which I find the book weak is harmonic functions, potential theory, and the applications of these to the calculus of resides. Consequently, the book is not “one-shop stopping” for all the mathematical techniques that an electrical engineer or physicist might require in his bag of tricks….

⭐While this booked is dated because it was written for the days of mechanical calculators, it contains a great deal of very useful material. His discussion of Chebyeshev Polynomials one of the best I seen. His discussion on telescoping of power series is one of the few available. He gives great insight into a host of numerical methods. A very valuable work for the computer age as well.

⭐Then this is the best book. Well, Hamming’s is also so good! For Fourier analysis, and the taming of the Gibbs phenomenon, go straight to Lanczos. He knew it all, and was one of the inventors of the fast Fourier transform. This book is in the class of Sommerfeld’s “Partial Differential Equations of Physics” and Lighthill’s “Fourier Analysis and Generalizaed Functions”. This is a very high compliment. Did you know he was also a first rate physicist, and a pioneer of quantum mechanics?

⭐Quel bonheur de trouver, à un prix modique, une version électronique de ce grand classique, vieux de plus de soixante ans. Même si certains passages sont obsolètes, ne serait-ce qu’en raison de l’importance prise par l’informatique, les citations toujours nombreuses de ce livre démontrent qu’étudiants et chercheurs ont toujours le plus grand intérêt à l’avoir sous le coude. On y trouve de multiples développements, qui n’ont pas d’équivalents dans la littérature d’aujourd’hui. Que l’on m’autorise à conclure par le souhait qu’Amazon publie une version Kindle de tous les traités de C. Lanczos et d’autres ouvrages anciens, célèbres, non seulement en anglais, mais aussi en allemand, français, … ? Merci d’avance !Fine. Good service.

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