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AuthorBrezinski, Claude. author
TitleHistory of Continued Fractions and Padรฉ Approximants [electronic resource] / by Claude Brezinski
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1991
Connect tohttp://dx.doi.org/10.1007/978-3-642-58169-4
Descript VIII, 551 p. online resource

SUMMARY

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 greatยญ 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 speakยญ 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 imยญ 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 tranยญ 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. Conยญ tinued fractions are also used in number theory, computer science, automata, electronics, etc ..


CONTENT

1 The Early Ages -- 1.1 Euclidโ{128}{153}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 Pellโ{128}{153}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 Padรฉ approximants -- 5 Maturity -- 5.1 Arithmetical continued fractions -- 5.2 Algebraic continued fractions -- 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 Padรฉ approximants -- 6.5 Extensions and applications -- Documents -- Document 1: Lโ{128}{153}algรจbre des gรฉomรจtres grecs -- Document 2: Histoire de lโ{128}{153}Acadรฉmie Royale des Sciences -- Document 3: Encyclopรฉdie (Supplรฉment) -- Document 4: Elementary Mathematics from an advanced standpoint -- Document 5: Sur quelques applications des fractions continues -- Document 6: Rapport sur un Mรฉmoire de M. Stieltjes -- Document 7: Correspondance dโ{128}{153}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


Mathematics Mathematical analysis Analysis (Mathematics) Numerical analysis Number theory Mathematics Numerical Analysis Number Theory Analysis



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