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Rating:  - Excellent Introduction to Complex Analysis
If you want to learn Complex Analysis, start with Schaum's Outlines. Then when you want to learn the methods and thinking of complex analysis, read this book. It's concise and gets the MAIN POINTS across in a friendly way.
Except for the topological stuff (they simplify things to avoid lengthy tedious discussion) this book is EXCELLENT. I disagree with the reviewers who said this book deals with things in too elementary a way. In fact it gives more general results and the REAL reasons behind complex numbers. Most importantly, it gives you a CONSISTENT FEEL for complex analysis techniques and concepts! For example, whereas most books treat a special case of the Riemann Principle of Removable Singularities where f is bounded. They use slightly tedious estimates of the Laurent coefficients to show that the terms with negative indices are all zero. This book simply shows that if lim (z-w)f(z) = 0 as z->w, then f(z) has a removable singularity, by appealing to the Schwartz Reflection Principle it proved earlier in the book. A more general result and gives a more integrated feeling for the theory.
ALL IN ALL A GREAT RESOURCE. After this, you can read Alfohrs, and then spcialized books on whatever. Lang is okay too but his results are not as general or intuitive as this book, and he uses power series constantly and is good for people who want a different perspective.
Rating:  - Good book
This book is excelent for a basic Complex Analysis course. It is very well writen, and the examples help you to understand the theorems. The book doesnt have to much solved examples, sometimes you need them.
I recommned to the complex Analysis book writen by Palka.
Rating:  - Hard to follow, not comprehensive enough
This book is disappointing, especially after encountering Newman's "Analytic Number Theory", which is a wonderful book. This book takes the readers on a concise, linear journey through Complex analysis to a few key theorems at the end, but does not do justice to the richness or diversity of the subject. This book will be especially lacking to students studying complex analysis for purposes related to applied mathematics.
The prose in the book is clear, but at times, as early as chapters 2 and 3, the equations are dense for an undergraduate text, with some steps less than obvious. There is a lack of motivation for the direction of development chosen in chapters 4-6, possibly a little of 2, 7, and 8 as well. Results are proven three or more times in cases of increasing generality. While this makes the theorems easy to follow, the redundancy may be confusing for a student studying the material for the first time. The authors do not provide much of a preview of what is to come, as I think authors of an undergraduate text should (and many, such as Gamelin, do).
This book is so small and compact that I question the authors' judgment in leaving out these various explanations--little would be lost and much gained by additional explanations. This makes me wonder what the intended audience is. I think anyone who is able to follow this book without trouble would also have no trouble following a more advanced and comprehensive book. There are a number of more advanced books that are actually much easier to follow. This brings me to my next comment:
This book leaves out a lot of important topics; it is far from comprehensive. There are not very many exercises either, and the exercises are mostly related to the material in simple ways.
For those studying complex analysis for the first time, I would recommend the Gamelin book over this one; its proofs are much easier to follow, it contains much more explanatory prose. It moves slower but it is much more comprehensive and covers more advanced material, and it is better suited to students with diverse interests and different backgrounds. I also recommend the Churchhill text as a straightforward book covering the basics. Advanced students might want to use the classic Ahlfors text.
Rating: - perhaps the best introduction to complex analysis
This is the book that really made me understand basic complex analysis. It doesn't try to give the most sophisticated or slickest presentation for experts. Instead, it gives a beautiful, concrete, down to earth explanations. The best feature is the applications. D. J. Newman is one of the world's great problem solvers, and this book includes numerous examples of how to use complex analysis to solve problems in surprising ways. Even in the more standard applications, such as summing series, the book gives many unusual examples. It concludes with Newman's proof of the prime number theorem, which is substantially shorter and clearer than many other proofs.
Rating: - Not enough for getting a complete perspective.
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).
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