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Series and products in the development of mathematics. In 2 volumes. 2nd updated edition of the book previously published in a single volume under the title ‘Sources in the development of mathematics’. (English) Zbl 1457.01002

Cambridge: Cambridge University Press (ISBN 978-1-108-70945-3/vol.1; 978-1-108-70937-8/vol.2; 978-1-108-70943-9/2-vol. set). xvii, 759 p./vol.1; xvii, 459 p./vol.2 (2021).
These two volumes form an extended version of the book [R. Roy, Sources in the development of mathematics. Infinite series and products from the fifteenth to the twenty-first century. Cambridge: Cambridge University Press (2011; Zbl 1228.01001)].
The first volume deals with power series in the Kerala school, sums of powers in the work of Faulhaber and Bernoulli, Wallis’s infinite products, the binomial theorem, applications to the rectification of curves and inequalities, results in calculus from Leibniz and Newton to Euler, the integration of rational functions, difference and differential equations, zeta functions, the Gamma function and Fourier series, the Euler-MacLaurin summation formula, hypergeometric series and orthogonal polynomials.
The second volume discusses more advanced material: \(q\)-series, partitions, Dirichlet \(L\)-series and the theorem of primes in arithmetic progression, the distribution of primes, the early phase of invariant theory, summability results, elliptic functions, transcendental numbers, value distribution theory, univalent functions and de Brange’s proof of the Bieberbach conjecture, and Weil’s work on the number of solutions of equations in finite fields.
The scope of the book is comparable to H. Dörrie’s [Unendliche Reihen. München: R. Oldenbourg Verlag (1951; Zbl 0045.33305)], which is unfortunately missing from the bibliography, but Roy’s book has a completely different character: Dörrie’s book is a textbook whereas Roy studies the original contributions of the masters with great clarity (and in modern notation).
As an example, we present the content of Chapter 4 on the binomial theorem. Roy locates the first known traces of this result in Pingala’s work on poetry from the second century BC; then he discusses contributions by Brahmagupta and Yang Hui before turning to Arabic scientists such as al-Karaji and Omar Khayyam. After Pascal, Newton formulated the binomial theorem for rational exponents; Roy explains the derivations of Newton’s result by Gregory, Landen, Euler, Cauchy, Abel, Harkness and Morley, and saves Eisenstein’s beautiful proof to vol. II since it is related to \(q\)-series. At the end of each chapter, Exercises and notes on the literature are given.
I cannot recommend these two volumes highly enough: to historians of mathematics as well as to every mathematician who is interested in history, analysis, combinatorics or number theory, and to all students of mathematics. In particular I recommend it as compulsary reading for everyone who thinks that studying books can be replaced by watching a couple of videos.

MSC:

01-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to history and biography
01A05 General histories, source books
26-03 History of real functions
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