Brezinski C. History of Continued Fractions and Padé Approximants

Berlin: Springer, 1991. — 559 p. — (Springer Series in Computational Mathematics 12). — ISBN 978-3-642-63488-8, 978-3-642-58169-4.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...The Early Ages
Euclid's algorithm
The square root
Indeterminate equations
History of notations
The First Steps
Ascending continued fractions
The birth of continued fractions
Miscellaneous contributions
Pell's equation
The Beginning of The Theory
Brouncker and Wallis
Huygens
Number theory
Golden Age
Euler
Lambert
Lagrange
Miscellaneous contributions
The birth of Pade approximants
Maturity
Arithmetical continued fractions
Algebraic properties
Arithmetic
Applications
Number theory
Convergence
Algebraic continued fractions
Expansion methods and properties
Examples and applications
Orthogonal polynomials
Convergence and analytic theory
Pade approximants
Varia
The Modern Times
Number theory
Set and probability theories
Convergence and analytic theory
Pade approximants
Extensions and applications
Appendix
Documents:
L'algebre des geometres grecs
Histoire de l'Academie Royale des Sciences
EncycIopedie (Supplement)
Elementary Mathematics from an advanced standpoint
Sur quelques applications des fractions continues
Rapport sur un Memoire de M. Stieltjes
Correspondance d'Hermite et de Stieltjes
Notice sur les travaux et titres
Note annexe sur les fractions continues
Scientific Bibliography
Works
Historical Bibliography
Name Index
Subject Index