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1991 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

History of Continued Fractions and Padé Approximants

verfasst von: Claude Brezinski

Verlag: Springer Berlin Heidelberg

Buchreihe : Springer Series in Computational Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

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 ...

Inhaltsverzeichnis

Frontmatter
Introduction
Abstract
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 greatest common divisor at least three centuries B.C. As it is often the case and like Monsieur Jourdain in Molière’s “le bourgeois gentilhomme” (who was speaking in prose though he did not know he was doing so), continued fractions were used for many centuries before their real discovery.
Claude Brezinski
Chapter 1. The Early Ages
Abstract
As I mentioned in the introduction, algorithms equivalent to the modern use of continued fractions were in use for many centuries before their real discovery. This chapter is devoted to these early attempts. The best known example is Euclid’s algorithm for the greatest common divisor of two integers, which leads to a terminating continued fraction. The approximate simplification of fractions (as practiced by the Greeks), is also related to this algorithm. The fundamental question of the irrationality of the square root of two was an important question for many years. The approximate computation of square roots led to some numerical methods which can be viewed as the ancestors of continued fractions. Another important problem related to astronomy and architecture is that of the solution of diophantine equations. The so-called Pell’s equation was also treated by the ancients (mostly by Indian mathematicians), who can be credited with the early discovery of algorithms analogous to continued fractions. The chapter will end with a short account on the history of the notations for continued fractions.
Claude Brezinski
Chapter 2. The First Steps
Abstract
The birth place of continued fractions is obviously the north of Italy, since the first attempt for a general definition of a continued fraction was made by Fibonacci. Considering this will give us the opportunity to discuss ascending continued fractions, which can be traced as far as the Rhind mathematical papyrus (about 1600 years B.C.). The next contribution was that of Bombelli, who in 1572 gave an algorithm for extracting the square root, completely equivalent to its expansion into an infinite continued fraction. But the real discoverer of continued fractions was Cataldi, who synthesized Bombelli’s method in 1613. After that we shall examine some applications to number theory, the most important of which is the challenge raised by Fermat to his contemporary mathematicians about Pell’s equation, which induced Brouncker and Wallis to the study of this problem.
Claude Brezinski
Chapter 3. The Beginning of the Theory
Abstract
After the first attempts of Bombelli and Cataldi, the theory of continued fractions was ready to be developed. The beginning of the theory was due mostly to Wallis, as we shall see in the next section. Another important contribution was by Huygens, who will be discussed with some other contributors.
Claude Brezinski
Chapter 4. Golden Age
Abstract
The 17th century saw the beginning of the theory of continued fractions. The 18th century was really their golden age. It was marked by three outstanding mathematicians, Euler, Lambert and Lagrange, all belonging to the Berlin Academy of Sciences which was reorganized by Frederic the second (Berlin, 24.1.1712—Berlin, 17.8.1786). The 18th century was also that of the birth of Padé approximants, which are connected with continued fractions and played (and are still playing) an important rôle in applications. The century ended with the publication of the first encyclopaedias and mathematical dictionaries, which had great importance in the spreading of continued fractions among the mathematicians, thus preparing the formidable explosion of the subject during the 19th century.
Claude Brezinski
Chapter 5. Maturity
Abstract
The nineteenth century can be said to be the popular period for continued fractions. The subject was known to every mathematician and even every scholar in many countries. An army of mathematicians, among them most of the famous ones, contributed to the development of continued fractions and in their application to various problems of mathematics and physics.
Claude Brezinski
Chapter 6. The Modern Times
Abstract
In this chapter, we shall follow the history of continued fractions and Padé approximants up to 1939. Of course the results which were obtained during that period are mostly the continuation of those proved during the nineteenth century and the border with the preceding chapter is mainly arbitrary. This is, in particular, true for the work of Padé himself, which extends from 1892 to 1907. However, since a division was necessary after so long a chapter, I have made one, but the reader interested in one particular subject is invited to jump across that border.
Claude Brezinski
Backmatter
Metadaten
Titel
History of Continued Fractions and Padé Approximants
verfasst von
Claude Brezinski
Copyright-Jahr
1991
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-58169-4
Print ISBN
978-3-642-63488-8
DOI
https://doi.org/10.1007/978-3-642-58169-4