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Complex Integration and Cauchy's Theorem

Dover Publications
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In the mid-19th century, the work of Cauchy, Riemann and Weierstrass created the foundations modern complex analysis. So when this slender volume was published in 1914, it was still a relatively youthful branch of mathematics — as was its juvenile companion, ‘analysis situs’ (now known as point set topology). Therefore, by means of this slender volume, G. N. Watson may have been the first person to provide an introduction to complex analysis in the English language.

Appearing as Number 15 in the ‘Cambridge Tracts in Mathematics and Mathematical Physics’, the book’s stated purpose was to present ‘those propositions which are employed in a rigorous proof of Cauchy’s theorem, together with a brief account of some of the applications of the theorem to the evaluation of definite integrals’.

With this aim, Watson begins with a chapter on relevant aspects of Poincaré’s ideas on the ‘analysis situs’ of R2. Then, instead of resorting to the methods of anti-differentiation, his approach to complex integration is based upon Riemann summation and mesh-fineness etc. Indeed, there is no general introduction to complex differentiation, and readers have to wait until the historical notes that form the last chapter to arrive at mention of the Cauchy-Riemann equations. Instead, Watson introduces analyticity from first principles in terms of an ε-δ definition, and the proof of the Cauchy-Goursat theorem is much lengthier and more difficult, when compared to modern equivalents.

Having achieved the goals suggested by the title of this book, later chapters focus upon the calculus of residues, the evaluation of real definite integrals, and series expansion of complex functions. Other results include change of variables in complex integrals, differentiation of an integral with regard to one of the limits and uniform differentiability.

Despite the familiarity of the aforementioned topics, this ‘tract’ would not now be regarded as an introductory textbook for modern usage. Some of the terminology has changed over the years, its focus is much narrower and the treatment is more challenging. On the other hand, it would certainly be of interest to those concerned with the history of complex analysis — particularly because Watson has based his approach on classic French textbooks, such as those by Goursat and de la Vallé Poussin.

Peter Ruane is retired from a career spent in the provision of courses for teachers of (primary and secondary) mathematics.

Date Received: 
Monday, September 10, 2012
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G. N. Watson
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P. N. Ruane

I. Analysis Situs
II. Complex Integration
III. Cauchy's Theorem
IV. Miscellaneous Theorems
V. The Calculus of Residues
VI. The Evaluation of Definite Integrals
VII. Expansions in Series
VIII. Historical Summary
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Sunday, October 21, 2012