History of Continued Fractions and Padé Approximants

555 Pages·1991·44.4 MB·English

by  Claude Brezinski (auth.)

Springer Series in
Com putational
Mathematics
12
Editorial Board
R. L. Graham, Murray Hill
J. Stoer, WOrzburg
R. Varga, Kent (Ohio)
Claude Brezinski
History of
Continued Fractions and
Pade Approximants
Springer-Verlag
Berlin Heidelberg GmbH
Claude Brezinski
Laboratoire d'Analyse Numerique et d'Optimisation
UFR IEEA-M3, Universite des Sciences et Techniques
de lilie Flandres-Artois
F-59655 Villeneuve d'Ascq Cedex, France
Mathematics Subject Classification (1980):
01 XX01, 1O XX03, 1O A32, 41 A03, 41 A21
ISBN 978-3-642-63488-8
Library 01 Congress Cataloging-in-Publication Data
Brezinski, Claude, 1941-
History 01 continued Iractions and Pade approximants / Claude Brezinski. p. cm.
(Springer series in computational mathematics; 12)
Includes bibliographical relerences.
ISBN 978-3-642-63488-8 ISBN 978-3-642-58169-4 (eBook)
DOI 10.1007/978-3-642-58169-4
1. Continued Iractions. 2. Pade approximant. I. Title. 11. Series
QA295.B79 1991 515'.243-dc20 90-9746 CIP
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© Springer-Verlag Berlin Heidelberg 1991
Originally published by Springer-Verlag Berlin Heidelberg New York in 1991
Softcover reprint ofthe hardcover 1st edition 1991
2141/3140-543210
a
En hommage Monsieur Noel Gastinel,
mon Maitre.
a
"Or pour contribuer la culture generale et tirer de l'enseignement
des sciences tout ce qu'il peut donner pour la formation de l'esprit,
rien ne saurait remplacer l'histoire des efforts passes, rendue vivante
par Ie contact avec la vie des grands savants et la lente evolution des
idees.
Par ce moyen seulement on peut preparer ceux qui continueront
1'reuvre de la science, leur donner Ie sens de son perpetuel mouvement
et de sa valeur humaine ...
Rien ne vaut d'aller aux sources, de se mettre en contact aussi
frequent et complet que possible avec ceux qui ont fait la science et
qui en ont Ie mieux represente l'aspect vivant ...
Les exemples precedents mont rent bien comment au point de
vue de I' enseignement comme au point de vue de la recherche scien
tifique, il est indispensable de ne pas oublier l'histoire des idees - et
concurremment celle des hommes, puisque c'est par eux qu'on eclaire
les idees. Rien de tel que de lire les reuvres des savants d'autrefois,
rien de tel que de vivre avec ceux qui sont contemporains pour
penetrer la pensee intime des uns et des autres."
Paul Langevin
"Avant tout, je pense que, si 1'on veut faire des progres en
mathematiques, il faut etudier les maltres et non les eleves."
Niels Henrik Abel
"It is an extremely useful thing to have knowledge of the true
origins of memorable discoveries, especially those that have been
found not by accident but by dint of meditation. It is not so much
that thereby history may attribute to each man his own discover
ies and others should be encouraged to earn like commendation, as
that the art of making discoveries should be extended by considering
noteworthy examples of it."
Gottfried Wilhelm Leibniz
"Study the past if you would divine the future."
Confucius
"To understand science it is necessary to know its history."
Auguste Comte
Contents
INTRODUCTION 1
CHAPTER 1
THE EARLY AGES 3
1.1 Euclid's algorithm 3
1.2 The square root . 9
1.3 Indeterminate equations 28
1.4 History of notations 44
CHAPTER 2
THE FIRST STEPS 51
2.1 Ascending continued fractions . 51
2.2 The birth of continued fractions 61
2.3 Miscellaneous contributions 70
2.4 Pell's equation . .. . . . . 72
CHAPTER 3
THE BEGINNING OF THE THEORY 77
3.1 Brouncker and Wallis 77
3.2 Huygens 86
3.3 Number theory 92
CHAPTER 4
GOLDEN AGE 97
4.1 Euler 97
4.2 Lambert 109
4.3 Lagrange 112
4.4 Miscellaneous contributions 125
4.5 The birth of Pade approximants 128
CHAPTER 5
MATURITY. 141
5.1 Arithmetical continued fractions 142
5.1.1 Algebraic properties . . . . . 142
5.1.2 Arithmetic 145
5.1.3 Applications 160
5.1.4 Number theory 171
5.1.5 Convergence 187
5.2 Algebraic continued fractions 190
5.2.1 Expansion methods and properties 190
5.2.2 Examples and applications 193
5.2.3 Orthogonal polynomials 213
5.2.4 Convergence and analytic theory 224
5.2.5 Pade approximants 236
5.3 Varia . . . .. 255
CHAPTER 6
THE MODERN TIMES 261
6.1 Number theory 261
6.2 Set and probability theories 272
6.3 Convergence and analytic theory 284
6.4 Pade approximants 292
6.5 Extensions and applications 301
APPENDIX . . . . . . . . . 313
DOCUMENTS . . . . . . . . 313
Document 1: L'algebre des geometres grecs 314
Document 2: Histoire de l'Academie Royale des Sciences 320
Document 3: EncycIopedie (Supplement) ...... 324
Document 4: Elementary Mathematics from an advanced
standpoint . . . . . . . . . . . . . . . 326
Document 5: Sur quelques applications des fractions continues 329
Document 6: Rapport sur un Memoire de M. Stieltjes . 335
Document 7: Correspondance d'Hermite et de Stieltjes 337
Document 8: Notice sur les travaux et titres . . . . 338
Document 9: Note annexe sur les fractions continues 345
SCIENTIFIC BIBLIOGRAPHY . 347
WORKS . . . . . .. .. .. 463
HISTORICAL BIBLIOGRAPHY 469
NAME INDEX 493
SUBJECT INDEX . . . . . . . 547
INTRODUCTION
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 in 1882, an open problem for more than two thousand years,
11'
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 ...
These are the reasons, together with my own longstanding interest in the
field, which induced me to try writing such a history. When I started this
work, I soon realized that I was not a historian, that I knew nothing about
the methods of history, and that it is in fact quite difficult to adequately
approximate a historian. Thus, this book is more a collection of facts and
references about continued fractions than their history in the modern sense of
the word, that is a profound analysis of the ancient mathematical works on
the subject. Moreover, historical research is very time consuming. One has to
spend day after day in libraries, looking for old books and journals which are
usually hidden in some cellar that almost nobody has visited for several years.
One also has to jump from one library to another to find the relevant material.
For all these reasons, and also because the history of continued fractions is a
very wide subject spanning more than twenty centuries, this work is certainly
far from complete and several important aspects of the subject are certainly
missing. However, I hope that this history will be attractive to researchers
working in the field, both from the cultural and also from the mathematical
point of view. Moreover, I would like it to be the starting point of some
serious historical work.
2
The book is mostly arranged chronologically although it was impossible
to strictly follow the chronological order. Each chapter has been divided
into sections dealing with a single subject. Thus, a reader interested in the
development of anyone of them can jump from one chapter to the next.
To help further studies, I have added as comprehensive as possible a bib
liography and biographical index. The bibliography has been divided into
mathematical references and sources of more historical nature (denoted by
[H. D.
Usually, the first time that a mathematician is quoted, his name is followed
by the dates and places of his birth and death (whenever they were known to
me). They are written as : day. month. year. This information has also been
gathered in the biographical index. The other dates following an author's
name in the text indicate a date of publication. Quotations of an author are
printed in italics. The appendix entitled "Documents" contains interesting
historical or mathematical citations that have not been directly included in
the text for length reasons.
This history ends with the first part of our century, that is in 1939. A
synthetic view of the actual developments of the theory can be found in the
introduction of H.S. Wall's book [H. 419] or in the more recent book of W.B.
Jones and W.J. Thron [H. 207].
The reader has been assumed to be familiar at least with the elementary
properties of continued fractions such as algebraic properties and some con
vergence results. The knowledge of some applications, for example those to
number theory, is also assumed. These properties and applications can be
found in any textbook on the subject such as the two previously mentioned
works. Some others are mentioned in the historical bibliography.
Finally, I would like to acknowledge the help of many very kind colleagues
who greatly helped me, either by their experience or by providing me with
some references. They are too numerous to be thanked individually, but I
certainly shall not forget their suggestions. I would also like to deeply thank
the staffs of the numerous libraries from which I extracted the information
contained here. I always found these librarians to be extremely helpful and
talented.
It is a great pleasure to thank very much Mrs. Fran~oise Tailly who typed
the first version of this book from the manuscript and Mrs. Gail E. Bostic who
produced the splendid final typing on a word processing machine. They both
did a tremendous and difficult work. Dr. T.s. Norfolk had the redoutable
privilege to correct my English. He did it with great skill and care. He must
be credited for the legibility of the book.
I would like to express my deepest gratitude to Professor Richard S. Varga.
Without his interest and his tireless efforts this book would never have been
published. It is my privilege to dedicate him the success of the book ... if any.
Chapter 1
THE EARLY AGES
As I mentioned in the introduction, algorithms equivalent to the modern use
of continued fractions were in use for many centuries before their real discov
ery. 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 mathemati
cians), 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.
1.1 EUCLID'S ALGORITHM
One of the best known algorithms, which is taught even in secondary schools,
is the algorithm for computing the greatest common divisor (g.c.d.) of two
integers. This algorithm, attributed to the Greek mathematician EUCLID
(c. 306 B.C. - c. 283 B.C.), leads to a terminating continued fraction [H. 363,
Vol. 2]. (c. means circa = around).
Let a and b be two positive integers and let a > b. We set
ro = a
rl = b
and then we compute the sequence of integers (rk) by
k = 0,1, ...