The Rise and Development of the Theory of Series up to the Early 1820s (eBook)

Giovanni Ferraro (Autor)

eBook Download: PDF

2007 | 2008
XVI, 392 Seiten
Springer New York (Verlag)
978-0-387-73468-2 (ISBN)

Infos

The manuscript gives a coherent and detailed account of the theory of series in the eighteenth and early nineteenth centuries. It provides in one place an account of many results that are generally to be found - if at all - scattered throughout the historical and textbook literature. It presents the subject from the viewpoint of the mathematicians of the period, and is careful to distinguish earlier conceptions from ones that prevail today.

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The theory of series in the 17th and 18th centuries poses several interesting problems to historians. Indeed, mathematicians of the time derived num- ous results that range from the binomial theorem to the Taylor formula, from the power series expansions of elementary functions to trigonometric series, from Stirling's series to series solution of di?erential equations, from theEuler-MaclaurinsummationformulatotheLagrangeinversiontheorem, from Laplace's theory of generating functions to the calculus of operations, etc. Most of these results were, however, derived using methods that would be found unacceptable today, thus, if we look back to the theory of series priortoCauchywithoutreconstructinginternalmotivationsandtheconc- tual background, it appears as a corpus of manipulative techniques lacking in rigor whose results seem to be the puzzling fruit of the mind of a - gician or diviner rather than the penetrating and complex work of great mathematicians. For this reason, in this monograph, not only do I describe the entire complex of 17th- and 18th-century procedures and results concerning series, but also I reconstruct the implicit and explicit principles upon which they are based, draw attention to the underlying philosophy, highlight competing approaches, and investigate the mathematical context where the series t- ory originated. My aim is to improve the understanding of the framework of 17th- and 18th-century mathematics and avoid trivializing the complexity of historical development by bringing it into line with modern concepts and views and by tacitly assuming that certain results belong, in some unpr- lematic sense, to a uni?ed theory that has come down to us today.

Inhalt

Preface 7
Contents 12
From the beginnings of the 17th century to about 1720: Convergence and formal manipulation 15
1 Series before the rise of the calculus 17
2 Geometrical quantities and series in Leibniz 39
3 The Bernoulli series and Leibniz’s analogy 59
4 Newton’s method of series 66
5 Jacob Bernoulli’s treatise on series 92
6 The Taylor series 99
7 Quantities and their representations 105
8 The formal-quantitative theory of series 126
9 The first appearance of divergent series 132
From the 1720s to the 1760s: The development of a more formal conception 142
10 De Moivre’s recurrent series and Bernoulli’s method 144
11 Acceleration of series and Stirling’s series 152
12 Maclaurin’s contribution 158
13 The young Euler between innovation and tradition 165
14 Euler’s derivation of the Euler–Maclaurin summation formula 180
15 On the sum of an asymptotic series 189
16 Infinite products and continued fractions 193
17 Series and number theory 201
18 Analysis after the 1740s 208
19 The formal concept of series 222
The theory of series after 1760: Successes and problems of the triumphant formalism 237
20 Lagrange inversion theorem 238
21 Toward the calculus of operations 243
22 Laplace’s calculus of generating functions 249
23 The problem of analytical representation of nonelementary quantities 255
24 Inexplicable functions 261
25 Integration and functions 267
26 Series and differential equations 270
27 Trigonometric series 278
28 Further developments of the formal theory of series 286
29 Attempts to introduce new transcendental func-tions 299
30 D’Alembert and Lagrange and the inequality technique 304
The decline of the formal theory of series 311
31 Fourier and Fourier series 314
32 Gauss and the hypergeometric series 322
33 Cauchy’s rejection of the 18th-century theory of series 345
References 361
Author Index 381
Subject Index 384

Produkt-Details

• Erscheint lt. Verlag: 20.12.2007
• Reihe/Serie: Sources and Studies in the History of Mathematics and Physical Sciences
• Reihe/Serie: Sources and Studies in the History of Mathematics and Physical Sciences
• Zusatzinfo: XVI, 392 p. 21 illus.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geschichte der Mathematik
• Themenwelt: Technik
• Schlagworte: Calculus • differential equation • Fourier series • Number Theory • Taylor series
• ISBN-10: 0-387-73468-6 / 0387734686
• ISBN-13: 978-0-387-73468-2 / 9780387734682