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History of the uniform convergence. Origins and developments of the concept in the analysis of the 19th century. With a foreword by Gert Schubring. (Geschichte der gleichmäßigen Konvergenz. Ursprünge und Entwicklungen des Begriffs in der Analysis des 19. Jahrhunderts.) (German) Zbl 1305.01006

Wiesbaden: Springer Spektrum (ISBN 978-3-658-05938-5/pbk; 978-3-658-05939-2/ebook). xv, 252 p. (2014).
The book contains probably the most thorough investigation into the history of the notion of uniform convergence and its development in the 19th century. After describing the methodology used and some general remarks, the investigation starts with Cauchy and his famous Theorem of Sums on the summation of series. Then the reactions of Cauchy’s contemporaries are examined in detail. In the fourth chapter, the author discusses historiographic problems with Cauchy’s famous theorem in examining the interpretations of Spalt and Laugwitz as well as of Grattan-Guinness and Bottazzini. The notion of uniform convergence did not come spontaneously but was ‘in the air’ somehow around 1846/47. Seidel, Stokes, Björling and others are mentioned and their contributions are discussed. Then the lectures given by Weierstraßare in the focus and the development of uniform convergence in the course of some years of his research and teaching is examined. Starting from Weierstraß, the reception of his ideas is discussed and the role of Schwarz, Du Bois-Reymond, Heine, Harnack, and Stolz are looked at. An outlook is given concerning mainly the reception of the idea of uniformity. In several appendices, transcriptions and translations of some letters and treatises are given. The book ends with a bibliography and an index.
The book contains an important step towards an understanding of the concept of uniformity on a wider scale. It is very readable and gives a wealth of information. The book is written in German and the publisher would have done better if a larger format would have been chosen to avoid the use of a magnifying glass at some points.

MSC:

01-02 Research exposition (monographs, survey articles) pertaining to history and biography
01A55 History of mathematics in the 19th century
26-03 History of real functions

Biographic References:

Schubring, Gert
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