Complex Analysis [Paperback]

Serge Lang (Author)

Book Description

Publication Date: June 13, 2008 | ISBN-10: 3540780599 | ISBN-13: 978-3540780595

This is a new, revised third edition of Serge Lang's Complex Analysis. The first part of the book covers the basic material of complex analysis, and 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 in other texts, and the proofs using these methods often shed more light on the results than the standard proofs do. The first part of Complex Analysis is suitable for an introductory course on the undergraduate level, and the additional topics covered in the second part give the instructor of a graduate course a great deal of flexibility in structuring a more advanced course.

--This text refers to the Hardcover edition.

Product Details

• Paperback: 476 pages
• Publisher: Springer (June 13, 2008)
• Language: English
• ISBN-10: 3540780599
• ISBN-13: 978-3540780595
• Product Dimensions: 9.1 x 6.1 x 1 inches
• Shipping Weight: 1.6 pounds (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars  See all reviews (4 customer reviews)
• Amazon Best Sellers Rank: #800,430 in Books (See Top 100 in Books)

More About the Author

Discover books, learn about writers, read author blogs, and more.

Customer Reviews

Most Helpful Customer Reviews

17 of 34 people found the following review helpful:

3.0 out of 5 stars Not enough for getting a complete perspective., June 21, 2000

By 

Bernardo Vargas - See all my reviews

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

My comment refers to the third edition of this book, but I don't think the fourth could be much better.

First of all, this title shouldn't be included in the "Graduate Texts in Mathematics" series because the material it covers is covered in introductory undergraduate courses. Second, eventhough the author made a great effort to include as much topics as he could, the treatment of most of them is highly old-fashioned. I mean, he pays no attention to the most recent and elegant refinements of the basic theory, so the student is not immediately able to understand the real important ideas behind the subject. For example, nowadays the proof of the Cauchy integral formula is presented as a more ar less easy corollary of the general Stokes theorem. The Cauchy integral theorem is also obtained easily following the same fashion. Incredibly, the author explores this line in one appendix, but not well done, and apparently he doesn't realize that there is the key idea.

Also, keeping in mind that holomorphic functions are harmonic, most of the important results for holomorphic functions should follow at once from the corresponding ones for harmonic functions, but this old-fashioned texts don't take this remarkable important feature of complex analysis into account, making the treatment innecessarily complicated and leading the student to misunderstand both complex and harmonic analysis. Eventhough the book includes a whole chapter on harmonic functions, the author doesn't use their power as he should.

I'm afraid there are few famous introductory texts that I would suggest for first-timers. The best of them is Markushevitch, unfortunately out of print.

There is also another serious drawback: The author pays no attention at all to boundary value problems and therefore to the Cauchy-type integral, maybe the most important tool of complex analysis. The Hilbert transform is also not present.

If you have the opportunity take a look at Muskhelishvili's "Singular Integral Equations" and Gakhov's "Boundary Value Problems" and then you will understand my point.

Lang's book could be used as a companion text and as a reference for introductory courses. It's got some interestig excercises.

Its contents are: Complex Nubers and Functions; Power Series; Cauchy's Theorem, First Part; Winding Numbers and Cauchy's Theorem; Applications of Cauchy's Integral Formula; Calculus of Residues; Conformal Mappings; Harmonic Functions; Schwartz Reflection; The Riemann Mapping Theorem; Analytic Continuation Along Curves; Applications of the Maximum Principle and jensen's Formula; Entire and Meromorphic Functions; Elliptic Fuctions; The Gamma and Zeta Functions; The Prime number Theorem; Appendices.

Please take a look to the rest of my reviews (just click on my name above).

0 of 5 people found the following review helpful:

3.0 out of 5 stars Technical Language, December 11, 2010

By 

M. Jackson - See all my reviews
(REAL NAME)   

Amazon Verified Purchase

This review is from: Complex Analysis (Paperback)

I bought this book for the good name of the publisher and because it was a cheaper book they put out. I was saddened that within only a couple of pages the book became technical with how to write functions for instance that I personally am not used to. I am not a Math Major, only Computer Engineer. I was able to do much of the math already regard to these pages, but I could still not understand the author due to his way of writing functions. For instance he commonly uses the "mapping to" style if I recall, which is not a big deal, but it gets a little more annoying as time goes on. Perhaps with more time I could get through it, but my patience is much shorter just trying to understand what the author is saying besides understanding the math.

0 of 5 people found the following review helpful:

5.0 out of 5 stars Nice complex book, May 2, 2010

By 

Olumuyiwa Oluwasanmi (Albuquerque NM) - See all my reviews
(REAL NAME)   

Amazon Verified Purchase

This review is from: Complex Analysis (Paperback)

This book is well written except the printing quality could be a little better!