Customer Reviews

Complex Analysis: In the Spirit of Lipman Bers (Graduate Texts in Mathematics)

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4 star:		(1)
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the main feature of this book is inscribed in its title: "in the spirit of Lipman Bers"; so, it is a testimony to their (the authors) beloved teacher; as a matter of fact, in the bibliography they point to one of Bers's prefered author: Konrad Knopp: he even translated one of Knoppp's five books on function theory (Funktionentheorie in german), adding that whenever Bers had to advise about exercice books on this subject, he told his students to work in those books. As a tribute to their past master, I must award 5 stars to the book's authors, but something has been nagging at me since I first opened the book; first of all, there is a chapter on the topology of H(D), i.e. the topological vector space of all holomorphic functions on an open Subset D of the complex plane (the topology being the so called "topology of uniform convergence on compact subsets", which is a fine idea, albeit it is taken from a to z in Cartan's book: "Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes" which has been translated in english as: "Elementary Theory of Analytic Functions of one or Several Complex Variables" (Dover editions); it is a sad thing to observe that no tribute at all is given to the old gentleman (Cartan is currently 104 years old...) although his book is indicated in the bibliography with the single mention: "begins with a treatment of formal power series..."; as a matter of facts, I feel that Cartan's choise of exposition is both straitforward and flawless: you get power series and the exponential function (together with pi's definition, complex logarithm and complex trigonometry) in 30 pages, Goursat's theorem and Cauchy's integral form within 70 pages (marginally better than in the currently reviewed book); this is my first point; then, towards the end of the book, one can find a proof of the celebrated Stirling formula; this time, it is taken from top to bottom out of Ahlfors's book: "complex Analysis", truly as magnificent a book as Cartan's and this time they aknowledge it; all the better..., I think that Ahlfors's book is really a great book too. Bluntly, the book contains less material than Cartan, is much more expensive (but less than Ahlfors) so why bother with it? On the other side, the authors have taken stock from the some of greatest specialists and writers in the field (from a classical point of view) which means that their work contains some outstanding mathematics. In the end, it is four stars as a tribute to Bers, Cartan, Ahlfors and Serge Lang who is said to have traded a lot of ideas with Bers while he was writing his own book on complex analysis
(guess what: from page 37 to 46, Lang begins his chapter on power series
with formal power series (shades of his french background?).

This is an excellent first course on complex analysis. It develops the subject's beautiful geometric ideas in a very smooth and intuitive way, making learning easier on students. The book is also written in theorem/proof format (all the main results are written this way), which greatly helps when using the book as a reference (in contrast to Ahlfor's Complex Analysis, for example, where at times it is necessary to skip back a few pages to fully understand a certain result or concept). In all, I think this is a great book to have to either study independently the subject, use it for a first course or complementary material, or just as a reference.