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Computational Techniques for the Summation of Series is a text on the representation of series in closed form. The book presents a unified treatment of summation of sums and series using function theoretic methods. A technique is developed based on residue theory that is useful for the summation of series of both Hypergeometric and Non-Hypergeometric type. The theory is supported by a large number of examples. The book is both a blending of continuous and discrete mathematics and, in addition to its theoretical base; it also places many of the examples in an applicable setting. This text is excellent as a textbook or reference book for a senior or graduate level course on the subject, as well as a reference for researchers in mathematics, engineering and related fields.
Table des matières
1. Some Methods for closed form Representation.- 1 Some Methods.- 2 A Tree Search Sum and Some Relations.- 2. Non-Hypergeometric Summation.- 1 Introduction.- 2 Method.- 3 Burmann's Theorem and Application.- 4 Differentiation and Integration.- 5 Forcing Terms.- 6 Multiple Delays, Mixed and Neutral Equations.- 7 Bruwier Series.- 8 Teletraffic Example.- 9 Neutron Behaviour Example.- 10 A Renewal Example.- 11 Ruin Problems in Compound Poisson Processes.- 12 A Grazing System.- 13 Zeros of the Transcendental Equation.- 14 Numerical Examples.- 15 Euler'sWork.- 16 Jensen's Work.- 17 Ramanujan's Question.- 18 Cohen's Modification and Extension.- 19 Conolly's Problem.- 3. Bürmann's Theorem.- 1 Introduction.- 2 Bürmann's Theorem and Proof.- 3 Convergence Region.- 4. Binomial type Sums.- 1 Introduction.- 2 Problem Statement.- 3 A Recurrence Relation.- 4 Relations Between Gk (m) and Fk+1 (m).- 5. Generalization of the Euler Sum.- 1 Introduction.- 2 1-Dominant Zero.- 3 The K-Dominant Zeros Case.- 6.Hypergeometric Summation: Fibonacci and Related Series.- 1 Introduction.- 2 The Difference-Delay System.- 3 The Infinite Sum.- 4 The Lagrange Form.- 5 Central Binomial Coefficients.- 6 Fibonacci, Related Polynomials and Products.- 7 Functional Forms.- 7. Sums and Products of Binomial Type.- 1 Introduction.- 2 Technique.- 3 Multiple Zeros.- 4 More Sums.- 5 Other Forcing Terms.- 8. Sums of Binomial Variation.- 1 Introduction.- 2 One Dominant Zero.- 3 Multiple Dominant Zeros.- 4 Zeros.- 5 Non-zero Forcing Terms.- References.- About the Author.
Résumé
Computational Techniques for the Summation of Series is a text on the representation of series in closed form. The book presents a unified treatment of summation of sums and series using function theoretic methods. A technique is developed based on residue theory that is useful for the summation of series of both Hypergeometric and Non-Hypergeometric type. The theory is supported by a large number of examples. The book is both a blending of continuous and discrete mathematics and, in addition to its theoretical base; it also places many of the examples in an applicable setting. This text is excellent as a textbook or reference book for a senior or graduate level course on the subject, as well as a reference for researchers in mathematics, engineering and related fields.
Texte suppl.
From the reviews:
"This book collects in one volume the author’s considerable results in the area of the summation of series and their representation in closed form, and details the techniques by which they have been obtained. … the calculations are given in plenty of detail, and closely related work which has appeared in a variety of places is conveniently collected together. That the author passes on his extensive knowledge of the literature of results for series will also be valued by the interested scholar." (Katherine Seaton, The Australian Mathematical Society Gazette, 32:1, 2005)
"The author presents an unified treatment of summation of sums and series using function theoretic methods. … No book of this type exists which attempts to give a link, by developing a comprehensive method, between non-hypergeometric and hypergeometric summation. … This book is intended for use in the fields of applied mathematics, analysis, non-hypergeometric and hypergeometric summation, summation of series and automated techniques." (Antonio López-Carmona, Zentralblatt MATH, Vol. 1059 (10), 2005)
Commentaire
From the reviews:
"This book collects in one volume the author's considerable results in the area of the summation of series and their representation in closed form, and details the techniques by which they have been obtained. ... the calculations are given in plenty of detail, and closely related work which has appeared in a variety of places is conveniently collected together. That the author passes on his extensive knowledge of the literature of results for series will also be valued by the interested scholar." (Katherine Seaton, The Australian Mathematical Society Gazette, 32:1, 2005)
"The author presents an unified treatment of summation of sums and series using function theoretic methods. ... No book of this type exists which attempts to give a link, by developing a comprehensive method, between non-hypergeometric and hypergeometric summation. ... This book is intended for use in the fields of applied mathematics, analysis, non-hypergeometric and hypergeometric summation, summation of series and automated techniques." (Antonio López-Carmona, Zentralblatt MATH, Vol. 1059 (10), 2005)
