Special functions, which include the trigonometric functions, have been used for centuries. Their role in the solution of differential equations was exploited by Newton and Leibniz, and the subject of special functions has been in continuous development ever since. In just the past thirty years several new special functions and applications have been discovered. This treatise presents an overview of the area of special functions, focusing primarily on the hypergeometric functions and the associated hypergeometric series. It includes both important historical results and recent developments and shows how these arise from several areas of mathematics and mathematical physics. Particular emphasis is placed on formulas that can be used in computation. The book begins with a thorough treatment of the gamma and beta functions that are essential to understanding hypergeometric functions. Later chapters discuss Bessel functions, orthogonal polynomials and transformations, the Selberg integral and its applications, spherical harmonics, q-series, partitions, and Bailey chains. This clear, authoritative work will be a lasting reference for students and researchers in number theory, algebra, combinatorics, differential equations, applied mathematics, mathematical computing, and mathematical physics.
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Editorial Reviews
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"...the authors demonstrate a superb familiarity with the historical roots of their subject...All of the chapters are beautifully written...wonderful historical insights...Special Functions will certainly emerge as the chief textbook and reference on special functions for the next several years...This book joins F. W. J. Olver's Asymptotics and Special Functions, first published in 1974 [Academic Press, New York; MR 55 #8655], as the only general books on special functions during the past three decades that belong 'in the Hobbs class,' to quote G. H. Hardy." Mathematical Reviews
"The book is packed with brief, challenging superveniences that make it a browser's delight. One of the delightful features of this book is how the sense of history, of mathematics being created and savored, informs the text. This is a splendid work, and I predict that it will be a bestseller as well." Bulletin of the American Mathematical Society
"The book is packed with brief, challenging superveniences that make it a browser's delight. One of the delightful features of this book is how the sense of history, of mathematics being created and savored, informs the text. This is a splendid work, and I predict that it will be a bestseller as well." Bulletin of the American Mathematical Society
Book Description
Special functions, which include the trigonometric functions, have been used for centuries. Their role in the solution of differential equations was exploited by Newton and Leibniz, and the subject of special functions has been in continuous development ever since. This treatise presents an overview of the area of special functions, focusing on the hypergeometric functions and the associated hypergeometric series. It includes both important historical results and recent developments and shows how these arise from several areas of mathematics and mathematical physics. Particular emphasis is placed on formulas that can be used in computation.This clear, authoritative work will be a lasting reference for students and researchers in number theory, algebra, combinatorics, differential equations, mathematical computing, and mathematical physics.
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Most Helpful Customer Reviews
28 of 28 people found the following review helpful:
5.0 out of 5 stars
Encyclopedic and highly readable book on Special Functions,
By A Customer
This review is from: Special Functions (Encyclopedia of Mathematics and its Applications) (Hardcover)
This is one of the best books on Mathematics I have read recently;reading it gave me pleasure similar to I had felt while reading classics such as Hardy's Pure Mathematics and Titchmarsh's Theory of Functions about 30 years ago. Starting with a thorough and encyclopedic treatment of Gamma Functions , the book develops, rightly, the thesis of grandeur and ubiquitousness of Hypergeometric Functions,to then cover some of the most important special functions known today. And the beauty lies in precise,to the point, yet historically complete and 'multi-angled' as well as rigourous treatment of every topic covered, with a large number of exercises of nontrivial nature dealing successfully with details which probably could not be included in the main text because of desire to keep the size of the book within reasonable limits. Nonetheless, the authors have delightfully and almost magically been able to cover a lot of ground in a mere 600 odd pages.
6 of 6 people found the following review helpful:
5.0 out of 5 stars
A Modern Whittaker and Watson, Buy It,
By MathGeek741 "jackgoodman2" (Maryland, USA) - See all my reviews
This review is from: Special Functions (Encyclopedia of Mathematics and its Applications) (Paperback)
This book is great. It is the best overview I have ever seen of the primary special functions, as seen from a modern viewpoint. Buy it and you will spend many happy hours reading the theorems it contains, and doing the excercizes at the end of each chapter.
2 of 2 people found the following review helpful:
5.0 out of 5 stars
Encyclopedic and highly readable book on Special Functions,
By A Customer
This review is from: Special Functions (Encyclopedia of Mathematics and its Applications) (Hardcover)
This is one of the best books on Mathematics I have read recently;reading it gave me pleasure similar to I had felt while reading classics such as Hardy's Pure Mathematics and Titchmarsh's Theory of Functions about 30 years ago. Starting with a thorough and encyclopedic treatment of Gamma Functions , the book develops, rightly, the thesis of grandeur and ubiquitousness of Hypergeometric Functions,to then cover some of the most important special functions known today. And the beauty lies in precise,to the point, yet historically complete and 'multi-angled' as well as rigourous treatment of every topic covered, with a large number of exercises of nontrivial nature dealing successfully with details which probably could not be included in the main text because of desire to keep the size of the book within reasonable limits. Nonetheless, the authors have delightfully and almost magically been able to cover a lot of ground in a mere 600 odd pages.
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Inside This Book
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First Sentence:
Euler discovered the gamma function, (x), when he extended the domain of the factorial function. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
confluent equation, linearization formula, hypergeometric representation, ultraspherical polynomials, bilateral series, hypergeometric orthogonal polynomials, contiguous relations, very well poised, beta integral, master theorem, hypergeometric identities, pentagonal number theorem, triple product identity, hypergeometric equation, denominator parameters, reflection formula, partition analysis, hypergeometric series, quadratic transformations, integral analog, regular singularities, hypergeometric differential equation, duplication formula, kernel polynomials, harmonic polynomials Key Phrases - Capitalized Phrases (CAPs): (learn more)
Master Theorem, Hypergeometric Transformations, Proof First, Kummer's Fourier Expansion of Log, Prove Lemma New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Euler discovered the gamma function, (x), when he extended the domain of the factorial function. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
confluent equation, linearization formula, hypergeometric representation, ultraspherical polynomials, bilateral series, hypergeometric orthogonal polynomials, contiguous relations, very well poised, beta integral, master theorem, hypergeometric identities, pentagonal number theorem, triple product identity, hypergeometric equation, denominator parameters, reflection formula, partition analysis, hypergeometric series, quadratic transformations, integral analog, regular singularities, hypergeometric differential equation, duplication formula, kernel polynomials, harmonic polynomials Key Phrases - Capitalized Phrases (CAPs): (learn more)
Master Theorem, Hypergeometric Transformations, Proof First, Kummer's Fourier Expansion of Log, Prove Lemma New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 47
books:
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100
books
cite this book:
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- Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications) by George Gasper in Front Matter, and page 520
- Computation with Finitely Presented Groups (Encyclopedia of Mathematics and its Applications) by Charles C. Sims in Front Matter
- Stopping Times and Directed Processes (Encyclopedia of Mathematics and its Applications) by G. A. Edgar in Front Matter
- Handbook of Categorical Algebra: Volume 2, Categories and Structures (Encyclopedia of Mathematics and its Applications) (v. 2) by Francis Borceux in Front Matter
- Geometric Applications of Fourier Series and Spherical Harmonics (Encyclopedia of Mathematics and its Applications) by Helmut Groemer in Front Matter
See all 47 books this book cites
- Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (Developments in Mathematics) by Frank G. Garvan on 20 pages
- An Introduction to Basic Fourier Series (Developments in Mathematics) by S.K. Suslov on 10 pages
- Representation of Lie Groups and Special Functions: Recent Advances (Mathematics and Its Applications (closed)) by N.Ja. Vilenkin on 6 pages
- Fourier Analysis on Finite Groups and Applications (London Mathematical Society Student Texts) by Audrey Terras in Back Matter (1), Back Matter (2), and Back Matter (3)
- Differential Equations and Asymptotic Theory in Mathematical Physics: Wuhan University, Hubei China, 20-29 October 2003 (Series in Analysis) by Chen Hua on page 40, page 223, and page 238
See all 100 books citing this book
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