History of Continued Fractions and Padé Approximants

ISBN-13: 9783642634888 / Angielski / Miękka / 2012 / 551 str.

Claude Brezinski

Opis

The history of continued fractions is certainly one of the longest among those of mathematical concepts, since it begins with Euclid's algorithm for the great- est common divisor at least three centuries B.C. As it is often the case and like Monsieur Jourdain in Moliere's "Ie bourgeois gentilhomme" (who was speak- ing in prose though he did not know he was doing so), continued fractions were used for many centuries before their real discovery. The history of continued fractions and Pade approximants is also quite im- portant, since they played a leading role in the development of some branches of mathematics. For example, they were the basis for the proof of the tran- scendence of 11' in 1882, an open problem for more than two thousand years, and also for our modern spectral theory of operators. Actually they still are of great interest in many fields of pure and applied mathematics and in numerical analysis, where they provide computer approximations to special functions and are connected to some convergence acceleration methods. Con- tinued fractions are also used in number theory, computer science, automata, electronics, etc ...

Spis treści

1 The Early Ages.- 1.1 Euclid’s algorithm.- 1.2 The square root.- 1.3 Indeterminate equations.- 1.4 History of notations.- 2 The First Steps.- 2.1 Ascending continued fractions.- 2.2 The birth of continued fractions.- 2.3 Miscellaneous contributions.- 2.4 Pell’s equation.- 3 The Beginning of the Theory.- 3.1 Brouncker and Wallis.- 3.2 Huygens.- 3.3 Number theory.- 4 Golden Age.- 4.1 Euler.- 4.2 Lambert.- 4.3 Lagrange.- 4.4 Miscellaneous contributions.- 4.5 The birth of Padé approximants.- 5 Maturity.- 5.1 Arithmetical continued fractions.- 5.1.1 Algebraic properties.- 5.1.2 Arithmetic.- 5.1.3 Applications.- 5.1.4 Number theory.- 5.1.5 Convergence.- 5.2 Algebraic continued fractions.- 5.2.1 Expansion methods and properties.- 5.2.2 Examples and applications.- 5.2.3 Orthogonal polynomials.- 5.2.4 Convergence and analytic theory.- 5.2.5 Padé approximants.- 5.3 Varia.- 6 The Modern Times.- 6.1 Number theory.- 6.2 Set and probability theories.- 6.3 Convergence and analytic theory.- 6.4 Padé approximants.- 6.5 Extensions and applications.- Documents.- Document 1: L’algèbre des géomètres grecs.- Document 2: Histoire de l’Académie Royale des Sciences.- Document 3: Encyclopédie (Supplément).- Document 4: Elementary Mathematics from an advanced standpoint.- Document 5: Sur quelques applications des fractions continues.- Document 6: Rapport sur un Mémoire de M. Stieltjes.- Document 7: Correspondance d’Hermite et de Stieltjes.- Document 8: Notice sur les travaux et titres.- Document 9: Note annexe sur les fractions continues.- Scientific Bibliography.- Works.- Historical Bibliography.- Name Index.

O autorze

Continued fractions and Padé approximants have played an important role in the development of many branches of mathematics, such as spectral theory of operator the transcendence of +. The book is the first on the subject. It presents a chronological and complete history of continued fractions and Padé approximants. A bibliography of 2500 items and a biographical index of the 1500 persons quoted are included.