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Sources in the Development of Mathematics: Series and Products from the Fifteenth to the Twenty-first Century

Ranjan Roy
Cambridge University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Charles Ashbacher
, on

The modern mathematics student is often presented with a technique and told, “if the expression looks like this, then you will perform the following strategy to **** it.” Given the succinct presentation of what to do in this particular situation, the student rarely understands that the mathematical recipe they are following did not always exist. Furthermore, for most processes, there was a great deal of effort expended by many people as the first glimmerings of the operation were transformed from an untested step to a standard topic found in all textbooks covering the subject.

That is especially true of the foundational mainstay of the mathematics major, calculus. Nearly every book has a collection of differentiation and integration formulas printed on the insides of the covers. In between those covers the student steps from section to section, given a specific type of problem along with the algorithm to use in solving it.  

In each chapter of this book, a topic is selected and after some preliminary remarks that define the topic along with some historical background, there is a sequence of short sections highlighting work done on this topic by prominent mathematicians. For example, chapter 18 has the title “Symmetric Functions.” The section titles are then:

  • Euler’s Proofs of Newton’s Rule
  • Mclaurin’s Proof of Newton’s Rule
  • Waring’s Power Sum Formula
  • Gauss’s Fundamental Theorem of Symmetric Functions
  • Cauchy: Fundamental Theorem of Symmetric Functions
  • Cauchy: Elementary Symmetric Functions as Rational Functions of Odd Power Sums
  • Laguerre and Polya on Symmetric Functions
  • MacMahon’s Generalization of Waring’s Formula

Each chapter closes with a small set of exercises and notes on the literature. Forty-one different topics are dealt with in this way. Solutions to the exercises are not included.

The title of the books should really be “Resources” rather than “Sources,” because if you are an instructor with a desire to treat any of these topics in greater detail, the corresponding chapters may provide all the historical information that you will need.

Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.

1. Power series in fifteenth-century Kerala
2. Sums of powers of integers
3. Infinite product of Wallis
4. The binomial theorem
5. The rectification of curves
6. Inequalities
7. Geometric calculus
8. The calculus of Newton and Leibniz
9. De Analysi per Aequationes Infinitas
10. Finite differences: interpolation and quadrature
11. Series transformation by finite differences
12. The Taylor series
13. Integration of rational functions
14. Difference equations
15. Differential equations
16. Series and products for elementary functions
17. Solution of equations by radicals
18. Symmetric functions
19. Calculus of several variables
20. Algebraic analysis: the calculus of operations
21. Fourier series
22. Trigonometric series after 1830
23. The gamma function
24. The asymptotic series for ln Γ(x)
25. The Euler–Maclaurin summation formula
26. L-series
27. The hypergeometric series
28. Orthogonal polynomials
29. q-Series
30. Partitions
31. q-Series and q-orthogonal polynomials
32. Primes in arithmetic progressions
33. Distribution of primes: early results
34. Invariant theory: Cayley and Sylvester
35. Summability
36. Elliptic functions: eighteenth century
37. Elliptic functions: nineteenth century
38. Irrational and transcendental numbers
39. Value distribution theory
40. Univalent functions
41. Finite fields.