Elementary Theory of Analytic Functions of One or Several Complex Variables (Dover Books on Mathematics)

This little book contains roughly two third of all the materials typically covered in a graduate level intro. complex analysis.Unlike other typical American textbook on complex analysis, the author starts the book with elementary exposition of power series; so called Weierstrass's view point as the author calls it.At first it seems that the author is slow at getting to the major point of the theory, namely Cauchy's integral formula, because he spends a fair amount of pages on power series. However, this little exposition in the beginning of the book actually makes the book very efficient; unlike the American counterparts, which presents the Cauchy's viewpoint first, then introduces the power series, creates an obstruction on the flow of the logical presentation of the subject.This choice of arrangement of material makes the presentation of global and local version of Cauchy's formula very short and efficient. Also, its corollaries such as Laurent's series and residue theorem is also treated in very economical way without sacrificing any clarity.After the treatment of the aforementioned basic/major theory rest of the book proceeds in almost same manner as the standard American texts; product development, Mobius transformation(automorphism groups of the simple domains), Riemann mapping theorem, normal families, harmonic functions, etc.This book can be efficient and compact, because the author assumes that the readers are familiar with basic set topology(such as compactness, connectedness, topological group). Also curvilinear integration is treated in terms of differential forms, just as in Ahlfors' book. Since, differential form is not typically covered in standard American undergraduate courses, the reader who are not exposed to differential forms might be frustrated while using this book. Another possible obstruction when reading this book might be that you need some knowledge of summable families of a topological group; otherwise the reader won't completely understand all the materials on the power series.As you can expect from the very small size of this book, many other interesting and important subject matters are missing in this wonderful book; such as Runge's theorem, Jensen's formula, Hadamard's factorization theorem, covering spaces, homology version of Cauchy's theory and branched covering associated with proper holomorphic maps, just to name a few.So, it might not be wise to use this book as a replacement for a modern textbook on complex analysis. However, its efficient and extremely clear presentation on the basic material makes this book an indispensable reading for any math students IMHO.

When I was a student in mathematics, in 1966, the French version of this book was my prayer book. Between the old treatises who didn't care about set theory and the many modern and thick ones who appeared since that date and clearly explain all details without the "one sees that" in front of a drawing on the complex plane. For me this book is sentimental. A person interested in the evolution of how complex analysis is taught sould be interested too. P. M.

Well-written and clear, though the typesetting is a little dated. I would recommend this for a first course in graduate-level complex analysis. I am currently a third-year PhD student and I'm using it to fill in some gaps.

Good job.

It's very good.

This elegant little book by Henri Cantan covers both complex functions on one and several variables, and in that way (by the inclusion of several variables) it differs and stands out from most other books on complex variables at the beginning US-graduate level. It is a translation of an original French language version. I can recommend both the original and the translation. It is readable, and the exercises are plenty and excellent. Thanks to Dover, the translation is now readily available and cheap. Cartan's book starts with complex numbers, power series, and a review of the standard complex functions of one variable, e.g., the exponential, and the complex logarithm. Then follow holomorphic functions, Taylor and Laurent expansions, singularities, Cauchy's theorems, residues, analytic continuation, lots of examples, and beautifully illustrated. Included are also geometric topics, elementary complex geometry, Mobius transformations, automorphisms, transformation groups, differential forms, harmonic and analytic functions, Riemann surfaces, and infinite products, and a brief chapter on conformal mappings. The book is divided pretty evenly between one and several variables, with the second half being several variables. However each part can be read pretty much independently of the other. The book in its French edition was published first in the 1950ties, and the first English edition in 1963, and then starting 1995 reprinted by Dover. It is suitable as a text for a course or as a supplement in a standard beginning graduate course in complex function theory. While it contains the standard elements in such a course, we note that a systematic treatment of power series comes relatively late, in Chapter 10, beginning on page 195 (halfway into the book.) Some readers might want to begin with that. Of other Dover titles on the same subject, but considerably more elementary we recommend the books by Fisher, Volkovyskii et al, Silverman, Schwerdtfeger, and Flanigan. These books however only cover the case of a single variable. Review by Palle Jorgensen, August 5, 2006.

The book assumes a background in French Mathematical Placement test and is not written for those without good understanding of real analysis.

This is a masterful treatment of the subject, including power series, integration, the homotopy version of cauchy's theorem, residue calculus, differential equations, harmonic functions, and intro to riemann surfaces.i confess i do not recognize the book i read from the first review here however, as mine begins with formal power series and has rather little on several complex variables. I have both the 1963 hardback and the recent dover reprint and they are identical, but i have not seen the french version.It is sobering that after 2 years not one person has even reacted to this review. Are todays students totally clueless about what books to learn from? I have a phd in math and am an internationally known researcher in math. give yourself a break and give my suggestions a try. how much do you have to lose on a book of this price?if you doubt me i would recall that this was the book of choice at harvard in 1965 by the great john torrance tate.

I'm using this book to self study complex analysis, and I think it's great for the most part. The thing I like most about this book (and all of Henri Cartan's books) is that he very very clearly labels each section, definition,theorem, proposition,proof etc. So having this small headings makes it very clear to know what is going on. This is in sharp contrast to the book by Ahlfors, where very often key definitions are buried within a wall of text.I'm still only on chapter 2, but so far I love it! His choice of words is extremely clear and direct without several pages of detours into weird "motivation" (it's an amazing book, but I find it to be not as great as some of his other works, like his books on differential Calculus in Banach spaces/ differential forms).One thing to note is that Cartan already assumes familiarity with complex numbers, Fields, and basic topological notions (open, closed, compact, connected, continuity, homeomorphisms).

Great introductory textbook to analytic functions (that "elementary theory" well explains the contents of this book). The topics are presented in a clear way, with rigorous proofs, abundancy of theorems, and with the usual examples. It's quite short, but it covers all the main aspects of complex integration and analytic function theory.

An useful book for both under-grad and grad level,to have rigid knowledge on complex analysis.

It is a very good book!!

First of all, sorry for my digression.In my view, the uninitiated of complex analysis would understand the subject with ease by means of the so-called Weierstrass approach; namely starting off with power series at the early stage. It was ordinary in the prewar days in Europe, and such was even in Japan. For example, see Hurwitz's book, Knopp's, Titchmarsh's, or Saks and Zygmund's. What's more, the approach was arguably stood up for by the late Prof. Kiyoshi Oka's educational point of view.It's regrettable that the approach backed off from the mainstream in the postwar days, and instead the so-called real analysis approach based on harmonic functions and the like, superseded it. I'm not sure of the reason that such an approach dominates, more's the pity.This book by Cartan, though not the French original, clearly stands up for the Weierstrass approach, and so it is desirable at least to me. The book, however, deserves to be read by only students having grasp of general topology, and consequently emphasizes differential forms and the like. In a way,  Ahlfors's book , which is evidently contemptuous of the Weierstrass approach, may be kinder than Cartan's. Besides, I discovered a few of misprints, and only a few minor errors, which weren't present in the French original. For this reason, I rated the book as four-star. Ahlfors's book