Orthogonal polynomials and continued fractions : from Euler's point of view
著者
書誌事項
Orthogonal polynomials and continued fractions : from Euler's point of view
（Encyclopedia of mathematics and its applications / edited by G.-C. Rota, [122]）
Cambridge University Press, 2008
- : hardback
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注記
Includes bibliographical references (p. 466-474) and index
内容説明・目次
内容説明
Continued fractions, studied since Ancient Greece, only became a powerful tool in the eighteenth century, in the hands of the great mathematician Euler. This book tells how Euler introduced the idea of orthogonal polynomials and combined the two subjects, and how Brouncker's formula of 1655 can be derived from Euler's efforts in Special Functions and Orthogonal Polynomials. The most interesting applications of this work are discussed, including the great Markoff's Theorem on the Lagrange spectrum, Abel's Theorem on integration in finite terms, Chebyshev's Theory of Orthogonal Polynomials, and very recent advances in Orthogonal Polynomials on the unit circle. As continued fractions become more important again, in part due to their use in finding algorithms in approximation theory, this timely book revives the approach of Wallis, Brouncker and Euler and illustrates the continuing significance of their influence. A translation of Euler's famous paper 'Continued Fractions, Observation' is included as an Addendum.
目次
- Preface
- 1. Continued fractions: real numbers
- 2. Continued fractions: Algebra
- 3. Continued fractions: Analysis
- 4. Continued fractions: Euler
- 5. Continued fractions: Euler's Influence
- 6. P-fractions
- 7. Orthogonal polynomials
- 8. Orthogonal polynomials on the unite circle
- A1. Continued fractions, Observations
- Bibliography
- Index.
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