Complex Analysis (Graduate Texts in Mathematics) [Hardcover]

Serge Lang (Author)

Book Description

ISBN-10: 0387985921 | ISBN-13: 978-0387985923 | Publication Date: December 7, 1998 | Edition: 4th

Now in its fourth edition, the first part of this book is devoted to the basic material of complex analysis, while the second covers many special topics, such as the Riemann Mapping Theorem, the gamma function, and analytic continuation. Power series methods are used more systematically than is found in other texts, and the resulting proofs often shed more light on the results than the standard proofs. While the first part is suitable for an introductory course at undergraduate level, the additional topics covered in the second part give the instructor of a gradute course a great deal of flexibility in structuring a more advanced course.

Editorial Reviews

Review

"The very understandable style of explanation, which is typical for this author, makes the book valuable for both students and teachers." EMS Newsletter, Vol. 37, Sept. 2000 Fourth Edition S. Lang Complex Analysis "A highly recommendable book for a two semester course on complex analysis." —ZENTRALBLATTMATH

Product Details

• Hardcover: 485 pages
• Publisher: Springer; 4th edition (December 7, 1998)
• Language: English
• ISBN-10: 0387985921
• ISBN-13: 978-0387985923
• Product Dimensions: 9.2 x 6.4 x 1.1 inches
• Shipping Weight: 1.8 pounds (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars  See all reviews (7 customer reviews)
• Amazon Best Sellers Rank: #335,927 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

41 of 42 people found the following review helpful:

4.0 out of 5 stars Not TOO complex, July 21, 2000

By 

Mark Lee Ellis - See all my reviews

This review is from: Complex Analysis (Graduate Texts in Mathematics) (Hardcover)

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 principle

Chp 3: connected sets, integrals over paths, primitives ("antiderivatives"), local Cauchy thm, etc

Chp 4: winding numbers, global Cauchy Thm, Artin's proof

Chp 5: Applications of Cauchy's integral formula, Laurent series

Chp 6: Calculus of residues, evaluation of complex definate integrals, Fourier transforms, etc (fun stuff)

Chp 7: Comformal mapping, Schwarz lemma, analytic automorphisms of the Disc

Chp 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.

10 of 10 people found the following review helpful:

4.0 out of 5 stars Excellent, but inconsistent pace, unnecessary proofs in early chapters..., June 9, 2006

By 

Alexander C. Zorach "Alex Zorach" (Lancaster, PA) - See all my reviews
(REAL NAME)   

This review is from: Complex Analysis (Graduate Texts in Mathematics) (Hardcover)

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.

9 of 11 people found the following review helpful:

5.0 out of 5 stars sweet dude, December 19, 2004

By 

chicken head cut off "mcscientist" (Gainesville/Orsay France) - See all my reviews

This review is from: Complex Analysis (Graduate Texts in Mathematics) (Hardcover)

I dont like lang's algebra, ugrad linear algebra, or diff/riemannian manifolds books all that much, but i LOVED this one.

I think an undergrad with calculus and patience can read it.
there are characteristic lang-style things like research-oriented material, and he actually has examples. He covers topics towards the end of the book which arent common elsewhere, so i've never put it down. I am not a mathematician and I like this book. It's in one of my standard 8 books that I dont leave home without (4 physics 4 math)