# A Course Of Modern Analysis

01 Jan 1935-

TL;DR: The fifth edition of the book as discussed by the authors has been published in 2002 and it has been supplemented with more recent results and references where appropriate, but it still preserves the style and content of the original, but with some modifications.

Abstract: This classic work has been a unique resource for thousands of mathematicians, scientists and engineers since its first appearance in 1902. Never out of print, its continuing value lies in its thorough and exhaustive treatment of special functions of mathematical physics and the analysis of differential equations from which they emerge. The book also is of historical value as it was the first book in English to introduce the then modern methods of complex analysis. This fifth edition preserves the style and content of the original, but it has been supplemented with more recent results and references where appropriate. All the formulas have been checked and many corrections made. A complete bibliographical search has been conducted to present the references in modern form for ease of use. A new foreword by Professor S.J. Patterson sketches the circumstances of the book's genesis and explains the reasons for its longevity. A welcome addition to any mathematician's bookshelf, this will allow a whole new generation to experience the beauty contained in this text.

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01 Oct 1960

TL;DR: In this article, a generalization to two independent variables of Lagrange's expansion of an inverse function was given by Stieltjes and proved rigorously by Poincare.

Abstract: A generalization to two independent variables of Lagrange's expansion of an inverse function was given by Stieltjes and proved rigorously by Poincare. A new method of proof is given here that also provides a new and sometimes more convenient form of the generalization. The results are given for an arbitrary number of independent variables. Applications are pointed out to random branching processes, to queues with various types of customers, and to some enumeration problems.

179 citations

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TL;DR: Two general theorems that cover structures decomposed into elementary “components” in either the labelled or the unlabelled case are derived, by combining the continuity theorem for characteristic functions with singularity analysis techniques based on Hankel contours.

117 citations

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TL;DR: The existence of a non-renormalization theorem for Galileon scalar fields makes them especially attractive candidates for inflation as well as for late-time acceleration in braneworld models as mentioned in this paper.

117 citations

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TL;DR: In this paper, the derivation of dispersion relations is reviewed as well as the intermediate state analysis of Feynman diagrams in terms of Goldstone diagrams at finite temperatures, and applications to magnetic impurities, local two-level systems, intermediate valence models are given.

107 citations

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TL;DR: The results suggest that Patricia tries are very well balanced trees in the sense that a random shape of Patriciatries resembles the shape of complete trees that are ultimately balanced trees.

Abstract: The Patricia trie is a simple modification of a regular trie. By eliminating unary branching nodes, the Patricia achieves better performance than regular tries. However, the question is: how much on the average is the Patricia better? This paper offers a thorough answer to this question by considering some statistics of the number of nodes examined in a successful search and an unsuccessful search in the Patricia tries. It is shown that for the Patricia containing n records the average of the successful search length Sn asymptotically becomes 1/h1 · ln n + O(1), and the variance of Sn is either var Sn = c · ln n + 0(1) for an asymmetric Patricia or var Sn = 0(1) for a symmetric Patricia, where h1 is the entropy of the alphabet over which the Patricia is built and c is an explicit constant. Higher moments of Sn are also assessed. The number of nodes examined in an unsuccessful search Un is studied only for binary symmetric Patricia tries. We prove that the mth moment of the unsuccessful search length EUmn satisfies limn→∞EUmn/logm2n = 1, and the variance of Un is var Un = 0.87907. These results suggest that Patricia tries are very well balanced trees in the sense that a random shape of Patriciatries resembles the shape of complete trees that are ultimately balanced trees.

90 citations