COMPLEX ANALYSIS, 4TH EDITION by LANG SERGE (PDF)

Description

The present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. The first half, more or less, can be used for a one-semester course addressed to undergraduates. The second half can be used for a second semester, at either level. Somewhat more material has been included than can be covered at leisure in one or two terms, to give opportunities for the instructor to exercise individual taste, and to lead the course in whatever directions strikes the instructor’s fancy at the time as well as extra read­ ing material for students on their own. A large number of routine exer­ cises are included for the more standard portions, and a few harder exercises of striking theoretical interest are also included, but may be omitted in courses addressed to less advanced students. In some sense, I think the classical German prewar texts were the best (Hurwitz-Courant, Knopp, Bieberbach, etc. ) and I would recommend to anyone to lo

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Prior to Lang’s book under review several excellent texts existed but none of them, in my opinion, possessed, what I have come to understand about Lang’s books, the combination of rich pedagogical skill coupled with an excellent understanding about what research topics are – as of the writing of the book – currently important. I think Lang, whom I knew and had correspondence about this matter with, wrote to educate the next generation of mathematicians. So his books are not compendiums of everything that is known but rather guide books into the research frontiers. This book was a delight for me to read and reconnect with a subject, admittedly not my area of research while a professor at the University of California, Irvine, that I always found especially beautiful. Lang goes to gerat lengths to actually teach the subject in a detailed and leisurely way. Of course, given the series in which the book appears the reader is expected to make some effort as the material is not spoon-fed, but readers who make the effort will find themselves richly rewarded with a deep knowledge of parts of the theory of complex analysis.

⭐A well written treatment by the OG, what more can be said? Good reference to have and nice emphasis on analytic ideas and principles.

⭐good and fast

⭐Lang’s Complex Analysis is an very good text for anyone wanting to move beyond introductory complex analysis.

⭐good

⭐There are about as many opinions on this book as there are different books that Lang wrote, but there is a reason for this: this is one strange book, even among Lang’s.I will start out by saying what I like about this book: most of it. This book provides a lot of topological flavour to complex variables, which I find very helpful. To someone who thinks topologically, many of the proofs in this book will seem more intuitive than in other texts. This is particularly true when you get into more advanced material.Overall, the writing is very clear. Lang is excellent at providing motivation, especially as you get farther along in this book. Unlike some of his other books, he can’t be criticized as moving too fast in this book.Now the bad: the book starts out very slow, painfully so. It seems the first chunk of the book is aimed at teaching rigorous complex analysis to someone whose background in analysis is weak. Lang repeats all of the basic theorems about limits, differentiation, convergence, etc. in full detail. However, the material picks up eventually, and by the end of the book it’s moving fast enough that anyone who enjoyed the first part will have trouble understanding the later material. This book covers a lot more material than most undergrad books on the subject, so I suppose it lives up to the GTM title.Bottom line: I don’t like the choice or order of topics in initial chapters. Some of the “new” material specific to complex variables is mixed in with old results common to basic analysis on the real line. Anyone with a good background in analysis will be frustrated trying to find what they need to learn. Also, Lang confuses the logic of the subject by working with the terms “analytic” and “holomorphic” separately for a great deal of time before showing their equivalence. His definitions, terminology, and development don’t line up with many other authors, and he has not convinced me that his choice of development was justified…because most of the stuff I like in this book comes after the first few chapters. However, if you can get past these hurdles, you’ll find that this is a pretty great book that has a lot to offer.

⭐A person with absolutely no knowledge of complex numbers could begin with page one of this book. However, I think that some exposure to analysis is helpful before finishing the first chapter, but not necessary. I found this book easier to read & understand than some real analysis books, yet it helped me further understand real analysis in the process. I’m sure this is due to mere repetition of some of those concepts over a different field. As the author mentions in his foreword, the first half of the book can be used as an undergraduate text (Jr/Sn years) and the second half can also, but I would NOT have enjoyed it in undergraduate studies. I found it worthy of a first course in complex numbers at the graduate level. I especially liked it after studying real numbers. The placement of the chapter subject matter can be altered (to some degree) to ones liking. I think Lang has provided good examples & problems. There’s a solutions manual (by Rami Shakarchi) for this text somewhere.A brief discription of the chapters (some of them at least):Chp 1: basic definitions & operations, polar form, functions, limits, compact sets, differentiation, Cauchy-Riemann eqs, angles under holomorphic (“differentiable”) maps.Chp 2: formal & convergent power series, analytic functions, inverse & open mapping thms., local maximum modulus principleChp 3: connected sets, integrals over paths, primitives (“antiderivatives”), local Cauchy thm, etcChp 4: winding numbers, global Cauchy Thm, Artin’s proofChp 5: Applications of Cauchy’s integral formula, Laurent seriesChp 6: Calculus of residues, evaluation of complex definate integrals, Fourier transforms, etc (fun stuff)Chp 7: Comformal mapping, Schwarz lemma, analytic automorphisms of the DiscChp 8: Harmonic functions; Chp 9: Schwarz reflection; Chp 10: Riemann mapping theorem; (11): Analytic continuation along curves; (12) applications of Maximum Modulus Principle an Jensen’s Formula; (13) Entire & Meromorphic functions; (14) elliptic functions; (15) Gamma & Zeta functions; (16) The Prime Number Theorem; and a handy appendix.

⭐Springer is not that great publisher it usually was 20 years ago.Great mathematics text but crappy printing. Pages are glued notbond to the cover and the paper texture is like hygienic paper.You can compare in my pictures the paper and printing quality of myold copy of GTM 94 vs. the new GTM 103 I just bought.The cover is made of thick printed cardboard covered with a thin layerof plastic. The older covers where covered with a layer of good qualitycanvas before printed so they were very strong.I am wondering to go with the pirate copies as there is not to muchdifference once printed with my laser printer, hum still I can say myin-house printing quality is much better!Definitively, I will never buy any book from Springer now on.

⭐Use this book in conjunction with another text. Difficult to learn from fully without reference to other sources. Overall very good. Note that answers are not included at the back of the book, making it a pain to track down if you are fully understanding something.

⭐This book is very good for learning complex analysis, which contains several interesting exercises as well.

⭐Non credo ci sia molto da dire, il libro è molto specialistico. Chiaro e ben fatto, la manifattura è ottimaOk, tel qu’attendu. Merci.

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