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Ferraro, Giovanni

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

Ferraro, Giovanni - The Rise and Development of the Theory of Series up to the Early 1820s, ebook

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ISBN: 9780387734682
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Table of contents

Part I. From the beginnings of the 17th century to about 1720: Convergence and formal manipulation

1. Series before the rise of the calculus
Giovanni Ferraro

2. Geometrical quantities and series in Leibniz
Giovanni Ferraro

3. The Bernoulli series and Leibniz’s analogy
Giovanni Ferraro

4. Newton’s method of series
Giovanni Ferraro

5. Jacob Bernoulli’s treatise on series
Giovanni Ferraro

6. The Taylor series
Giovanni Ferraro

7. Quantities and their representations
Giovanni Ferraro

8. The formal-quantitative theory of series
Giovanni Ferraro

9. The first appearance of divergent series
Giovanni Ferraro

Part II. From the 1720s to the 1760s: The development of a more formal conception

10. De Moivre’s recurrent series and Bernoulli’s method
Giovanni Ferraro

11. Acceleration of series and Stirling’s series
Giovanni Ferraro

12. Maclaurin’s contribution
Giovanni Ferraro

13. The young Euler between innovation and tradition
Giovanni Ferraro

14. Euler’s derivation of the Euler–Maclaurin summation formula
Giovanni Ferraro

15. On the sum of an asymptotic series
Giovanni Ferraro

16. Infinite products and continued fractions
Giovanni Ferraro

17. Series and number theory
Giovanni Ferraro

18. Analysis after the 1740s
Giovanni Ferraro

19. The formal concept of series
Giovanni Ferraro

Part III. The theory of series after 1760: Successes and problems of the triumphant formalism

20. Lagrange inversion theorem
Giovanni Ferraro

21. Toward the calculus of operations
Giovanni Ferraro

22. Laplace’s calculus of generating functions
Giovanni Ferraro

23. The problem of analytical representation of nonelementary quantities
Giovanni Ferraro

24. Inexplicable functions
Giovanni Ferraro

25. Integration and functions
Giovanni Ferraro

26. Series and differential equations
Giovanni Ferraro

27. Trigonometric series
Giovanni Ferraro

28. Further developments of the formal theory of series
Giovanni Ferraro

29. Attempts to introduce new transcendental functions
Giovanni Ferraro

30. D’Alembert and Lagrange and the inequality technique
Giovanni Ferraro

Part IV. The decline of the formal theory of series

31. Fourier and Fourier series
Giovanni Ferraro

32. Gauss and the hypergeometric series
Giovanni Ferraro

33. Cauchy’s rejection of the 18th-century theory of series
Giovanni Ferraro

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Keywords: MATHEMATICS / General MAT000000

Author(s)
Publisher
Springer
Publication year
2008
Language
en
Edition
1
Category
Natural Sciences
Format
Ebook
eISBN (PDF)
9780387734682

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