This new and exciting historical book tells how Euler introduced the idea of orthogonal polynomials and how he combined them with continued fractions, as well as 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.
Author(s): Sergey Khrushchev
Series: Encyclopedia of Mathematics and its Applications
Publisher: CUP
Year: 2008
Language: English
Pages: 495
Cover......Page 1
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Contents ......Page 8
Preface ......Page 10
1.1 Historical background ......Page 18
1.2 Euler's theory of continued fractions ......Page 28
1.3 Rational approximations ......Page 34
1.4 Jean Bernoulli sequences ......Page 53
1.5 Markoff sequences ......Page 66
2.1 Euler's algorithm ......Page 88
2.2 Lagrange's theorem ......Page 98
2.3 Pell's equation ......Page 101
2.4 Equivalent irrationals ......Page 109
2.5 Markoff's theory ......Page 115
3.1 Convergence: elementary methods ......Page 140
3.2 Contribution of Brouncker and Wallis ......Page 148
3.3 Brouncker's method and the gamma function ......Page 163
4.1 Partial sums ......Page 175
4.2 Euler's version of Brouncker's method ......Page 180
4.3 An extension of Wallis’ formula ......Page 186
4.4 Wallis’ formula for sinusoidal spirals ......Page 191
4.5 An extension of Brouncker's formula ......Page 194
4.6 On the formation of continued fractions ......Page 197
4.7 Euler's differential method ......Page 200
4.8 Laplace transform of hyperbolic secant ......Page 208
4.9 Stieltjes’ continued fractions ......Page 211
4.10 Continued fraction of hyperbolic cotangent ......Page 216
4.11 Riccati's equation ......Page 223
5 Continued fractions: Euler's influence ......Page 245
5.1 Bauer-Muir-Perron theory ......Page 246
5.2 From Euler to Scott-Wall ......Page 249
5.3 The irrationality of ......Page 255
5.4 The parabola theorem ......Page 257
6.1 Laurent series ......Page 264
6.2 Convergents ......Page 270
6.3 Quadratic irrationals ......Page 275
6.4 Hypergeometric series ......Page 289
6.5 Stieltjes’ theory ......Page 302
7.1 Euler's problem ......Page 313
7.2 Quadrature formulas ......Page 315
7.3 Sturm's method ......Page 320
7.4 Chebyshev's approach to orthogonal polynomials ......Page 327
7.5 Examples of orthogonal polynomials ......Page 332
8.1 Orthogonal polynomials and continued fractions ......Page 339
8.2 The Gram-Schmidt algorithm ......Page 353
8.3 Szego's alternative ......Page 363
8.4 Erdos measures ......Page 373
8.5 The continuum of Schur parameters ......Page 377
8.6 Rakhmanov measures ......Page 381
8.7 Convergence of Schur's algorithm on T ......Page 385
8.8 Nevai's class ......Page 388
8.9 Inner functions and singular measures ......Page 397
8.10 Schur functions of smooth measures ......Page 405
8.11 Periodic measures ......Page 407
Appendix Continued fractions, observations L. Euler (1739) ......Page 443
References ......Page 483
Index ......Page 492
