History of Continued Fractions and Padé Approximants

by Claude Brezinski

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.

Genres: Mathematics

566 pages, Paperback

First published December 1, 1990