The authors’ aim here is to present a precise and concise treatment of those parts of complex analysis that should be familiar to every research mathematician. They follow a path in the tradition of Ahlfors and Bers by dedicating the book to a very precise goal: the statement and proof of the Fundamental Theorem for functions of one complex variable. They discuss the many equivalent ways of understanding the concept of analyticity, and offer a leisure exploration of interesting consequences and applications. Readers should have had undergraduate courses in advanced calculus, linear algebra, and some abstract algebra. No background in complex analysis is required.
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Book Description
Editorial Reviews
Review
From the reviews: "This is a fairly conventional text for a first course in complex analysis. It is an interesting mix of the concrete and the abstract, and of the formulaic and the geometric. It has good exercises … . It is nominally a graduate text (it is in Springer’s series of Graduate Texts in Mathematics) … . The book covers all the usual topics for a first course and includes a lot of advanced topics … ." (Allen Stenger, MathDL, February, 2008) "This book is based on the original courses of Complex Analysis that was delivered by the well-known American specialist and lecturer in Analysis, Professor Lipman Bers. … the book is written in a clear and easily readable manner. … this book is useful for all specialists and lecturers in Complex Analysis … and also all specialists who deal with applications of Complex Analysis. I also recommend this book to beginners who want to study Complex Analysis … ." (Peter Zabreiko, Zentralblatt MATH, Vol. 1139 (17), 2008)
From the Back Cover
This book is intended for a graduate course on complex analysis, also known as function theory. The main focus is the theory of complex-valued functions of a single complex variable. This theory is a prerequisite for the study of many current and rapidly developing areas of mathematics including the theory of several and infinitely many complex variables, the theory of groups, hyperbolic geometry and three-manifolds, and number theory. Complex analysis has connections and applications to many other subjects in mathematics and to other sciences. It is an area where the classic and the modern techniques meet and benefit from each other. This material should be part of the education of every practicing mathematician, and it will also be of interest to computer scientists, physicists, and engineers. The first part of the book is a study of the many equivalent ways of understanding the concept of analyticity. The many ways of formulating the concept of an analytic function are summarized in what is termed the Fundamental Theorem for functions of a complex variable. The organization of these conditions into a single unifying theorem with an emphasis on clarity and elegance is a hallmark of Lipman Bers's mathematical style. Here it provides a conceptual framework for results that are highly technical and often computational. The framework comes from an insight that, once articulated, will drive the subsequent mathematics and lead to new results. In the second part, the text proceeds to a leisurely exploration of interesting ramifications of the main concepts. The book covers most, if not all, of the material contained in Bers’s courses on first year complex analysis. In addition, topics of current interest such as zeros of holomorphic functions and the connection between hyperbolic geometry and complex analysis are explored.
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7 of 7 people found the following review helpful:
4.0 out of 5 stars
A tribute to old masters,
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This review is from: Complex Analysis: In the Spirit of Lipman Bers (Graduate Texts in Mathematics) (Hardcover)
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?).
1 of 1 people found the following review helpful:
5.0 out of 5 stars
Excellent,
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This review is from: Complex Analysis: In the Spirit of Lipman Bers (Graduate Texts in Mathematics) (Paperback)
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.
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Inside This Book
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Key Phrases - Statistically Improbable Phrases (SIPs):
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mean value property, residue theorem, more boundary points, continuous closed path, nonconstant holomorphic function, hyperbolic length, complex power series, conformal equivalence, differentiable path, holomorphic functions, extended complex plane Key Phrases - Capitalized Phrases (CAPs): (learn more)
Fundamental Theorem, Cauchy Integral Formula, Riemann Mapping Theorem, Euler's F-function Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Surprise Me!
mean value property, residue theorem, more boundary points, continuous closed path, nonconstant holomorphic function, hyperbolic length, complex power series, conformal equivalence, differentiable path, holomorphic functions, extended complex plane Key Phrases - Capitalized Phrases (CAPs): (learn more)
Fundamental Theorem, Cauchy Integral Formula, Riemann Mapping Theorem, Euler's F-function Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Surprise Me!
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