Cauchy and the Creation of Complex Function Theory [Hardcover]

Frank Smithies (Author)

Book Description

Publication Date: November 28, 1997 | ISBN-10: 052159278X | ISBN-13: 978-0521592789

In this book, Dr. Smithies analyzes the process through which Cauchy created the basic structure of complex analysis, describing first the eighteenth century background before proceeding to examine the stages of Cauchy's own work, culminating in the proof of the residue theorem and his work on expansions in power series. Smithies describes how Cauchy overcame difficulties including false starts and contradictions brought about by over-ambitious assumptions, as well as the improvements that came about as the subject developed in Cauchy's hands. Controversies associated with the birth of complex function theory are described in detail. Throughout, new light is thrown on Cauchy's thinking during this watershed period. This book is the first to make use of the whole spectrum of available original sources and will be recognized as the authoritative work on the creation of complex function theory.

Editorial Reviews

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"...this scholarly work is recommended for upper-division undergraduates through researchers and historians of mathematics." Choice

"This book is a welcome addition to the literature." Mathematical Reviews

Book Description

Dr Smithies' analysis of the process whereby Cauchy created the basic structure of complex analysis, begins by describing the 18th century background. He then proceeds to examine the stages of Cauchy's own work, culminating in the proof of the residue theorem. Controversies associated with the the birth of the subject are also considered in detail. Throughout, new light is thrown on Cauchy's thinking during this watershed period. This authoritative book is the first to make use of the whole spectrum of available original sources.

Product Details

• Hardcover: 228 pages
• Publisher: Cambridge University Press (November 28, 1997)
• Language: English
• ISBN-10: 052159278X
• ISBN-13: 978-0521592789
• Product Dimensions: 9.1 x 6.3 x 1 inches
• Shipping Weight: 1.1 pounds (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #3,578,456 in Books (See Top 100 in Books)

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5 of 5 people found the following review helpful:

4.0 out of 5 stars Great piece of Mathematical History!, December 17, 2001

By 

"jayjina" - See all my reviews

This review is from: Cauchy and the Creation of Complex Function Theory (Hardcover)

This is a great book on the life and Complex Variable Analysis work of Augustine Cauchy.

Smithies brings many of the quaint, yet pertinent pieces that comprise the rich field called Complex Variables together into a mathematical jigsaw. I for one was fascinated to see the link between the CR Equations and Green's Theorem.

Using copious examples, directly from Cauchy's published as well as unpublished notes, Smithies enlightens the reader of the genius of Cauchy. To the reader versed in modern representation of Complex Variable theory, Smithies insistence on sticking to Cauchy's formulation of complex functions in real terms and then using his method of separation may seem laborious at first, but it rewards the reader by giving deep insight into Cauchy's thought processes.

The book covers a wide field of Complex Variables from the formulation of Cauchy's Theorem, Cauchy's Integral Formula, the Residue Theorem and Integrals between imaginary limits.

Remarkable though his work was, the book shows the almost insular nature of Cauchy - by sometimes never publishing his results and being either unaware or disdainful of other results, e.g. d':Alembert, Gauss, Euler, etc.

A good book to read as an accompaniment to a standard text on Complex Variables (e.g. Ablowitz and Fokas). Rich in Mathematics and History and highly recommended!

1 of 1 people found the following review helpful:

4.0 out of 5 stars A field guide to Cauchy's function theory, March 25, 2006

By 

Viktor Blasjo - See all my reviews
(REAL NAME)   

This review is from: Cauchy and the Creation of Complex Function Theory (Hardcover)

In chapter 1 we look at what was done before Cauchy. Euler and Laplace and others saw that complex changes of variables was a useful technique for evaluating real integrals. This is rather mysterious and it was treated with some suspicion. Cauchy set out to justify and systematise such techniques in his 1814 memoir (chapter 2), and then he kept polishing his results over the next ten years (chapter 3). Then comes his watershed 1825 memoir (chapter 4). Cauchy has now realised that all of the above should be understood in the context of path integration in the complex plane. Here integration is largely determined by the poles, prompting a calculus of residues, which he develops over the next couple of years (chapter 5). Another area of classical analysis where the complex viewpoint proved essential was the convergence of series (chapter 6). Cauchy's starting point here was the Lagrange series, first employed fifty years earlier by Lagrange, e.g. in celestial mechanics, without regard for its dubious convergence properties.

Let me add some details about Cauchy's crucial 1814 paper which marked the transition from naive to systematic complex function theory and was to be the germ of all his later work in complex analysis. Laplace called the use of complex substitutions to evaluate real integrals "un moyen fécond de découvertes"---a fruitful method of discovery. But by way of justifying these methods he offered little but a vague appeal to "la généralité d'analyse"---the generality of analysis. Although his results were all correct, Poisson still found it worthwhile to rederive them by other methods, since, as he said, Laplace's reasoning was "une sort d'induction fondée sur le passage des quantités réelles aux imaginaires"---a sort of induction based on the passage from real to imaginary quantities. In a reply,Laplace agreed that the use of complex variables constituted "une analogie singulière"---a singular analogy---which "laissent toujours à désirer des démonstrations directes"---still left a desire for direct demonstrations---and he proceeded to offer some such demonstrations himself. As a result of this debate, Laplace assigned his young and ambitious protégé, Cauchy, the task of investigating the foundations of complex methods in integration.

The day after his 25th birthday, Cauchy presented his "Mémoire sur les intégrales définies," aiming to "établir le passage du réel à l'imaginaire sur une analyse directe et rigoureuse"---base the passage from the real to the imaginary on a direct and rigorous analysis. His basic tool for accomplishing this is "Cauchy's theorem." However, the form in which this theorem appears in his paper is very different from its modern statement. First of all, Cauchy never uses any geometrical language whatsoever in this paper. In particular, he never speaks of integrating along a path in the complex plane. Furthermore, he almost exclusively splits every complex expression into its real and imaginary parts and works with real integrals for each separately. In part these preferences can be explained by his stated goal of making complex methods rigorous. However, they are also a reflection of a very marked trend in French mathematics at the time, which emphasised an analytic-algebraic approach in all mathematics with almost chauvinistic determination.

Rather than having to do with path integration in the complex plane, Cauchy's "Cauchy's theorem" is the double integral of the Cauchy-Riemann equations. This is the fundamental idea of Cauchy's paper. This idea enables Cauchy to systematically derive all previous results established by complex methods, as well as many new results, in a way essentially equivalent to the modern textbook approach. And, crucially, he is able to do so with only minimal reference to complex numbers. He only really needs complex numbers to derive the Cauchy-Riemann equations, which is of course a simple matter of elementary algebra that no one can doubt the validity of. From there on everything is real. The Cauchy-Riemann equations are of course expressed in terms of the real and imaginary parts, and their double integrals are, in effect, the real an imaginary parts of Cauchy's theorem. This is the way in which Cauchy is able to defuse the issues of rigour that comes with liberal uses of complex numbers in integration.