The Real and the Complex: A History of Analysis in the 19th Century (Springer Undergraduate Mathematics Series) by Jeremy Gray (PDF)

Description

This book contains a history of real and complex analysis in the nineteenth century, from the work of Lagrange and Fourier to the origins of set theory and the modern foundations of analysis. It studies the works of many contributors including Gauss, Cauchy, Riemann, and Weierstrass.This book is unique owing to the treatment of real and complex analysis as overlapping, inter-related subjects, in keeping with how they were seen at the time. It is suitable as a course in the history of mathematics for students who have studied an introductory course in analysis, and will enrich any course in undergraduate real or complex analysis.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐When one learns analysis one starts with epsilon-delta arguments almost immediately and modern standards of rigor are forced on the reader from the beginning. Of course mathematics was not arrived at this way and the path to a rigorous foundation of analysis took centuries. The Real and the Complex takes a look at the development of analysis through the lives of many significant mathematicians and gives the reader a sense of the concepts they struggled with and eventually came to conclusions on. There is a huge amount of material in this book that can be read by a wide audience, though to truly follow much of it, I think much more knowledge is needed than what the author describes as the pre-requisites.The author focuses on a variety of topics and mathematicians but the main highlights were complex analysis and its foundations. Cauchy is of course at the center of much of the evolution of complex analysis and the use of the residue theorem is discussed along with the the complexities of differentiability and the algebraic vs geometric foundations of the subject. The author covers a lot in elliptic functions and elliptic integrals which are very much not a standard part of undergraduate analysis courses, so the work of Jacobi and Legendre are all quite interesting but is where the author’s descriptions of pre-requisites are highly dubious. The author covers the introduction of Riemann’s perspectives on complex functions and Riemann surfaces and the tensions it brought from Weierstrass, another key figure in the foundations of analysis. The author discusses personality aspects of each of the towering figures in the field and how they presented their works, or didn’t for that matter and some of the issues of priority. The foundations of analysis and epsilon delta methods are also discussed but subsequent to much of the work on complex analysis and Weierstrass becomes a major figure in the 2nd half of the book. The author very briefly covers some concepts in Topology, measure and integration theory and late 19th century analysis where nowhere differentiable functions were being struggled with. The author then ends with some set theory and cardinality of the sets.From a history of analysis standpoint, i think the books of Bressoud are superior, both for Lebesgue theory and for a first course on real analysis. I think for those who know these subjects and are looking for some history this is a great book, i think for one not knowing these subjects and having taken a first course in real analysis, good luck, you will be lost much of the time… I have enough familiarity with some of the topics as elliptic integrals come up in mathematical physics for example, but the authors presentation without any familiarity is not easy to learn much. All in all, this is a nice book to read and skim when it becomes hard to follow. There are quite a few typos, many from basic grammar which i would like to see edited out in any future editions. But a nice addition to a history of mathematics library.

⭐You could spend a lifetime enjoying this wonderful book.

⭐Interested in history and analysis/mathematics?You will find lots of information about mathematicians, their life, their work, their relationships. All about analysis in the 19th century.

⭐OK

⭐the idea of thsi book is good ;there is some small print errors;sometimes we would like more détails explanations;but the book is unique;

⭐

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