Complex Analysis

Rating:
Summary: a very good textbook
Review: In my viewpoint this book is one of the best complex analysis textbooks to date. It is succint and neat, without too many pages and too much content, while every facet of elementary complex analysis theory gets a chapter or two in it. It deals with power series first, then analytic fuctions, then singular pts and residue theorem, then conformal mapping. After these basic topic, it gives some futher theme like harmonic function and Riemann mapping theorem. And the last with some chapters, including a topic on proving prime number theorem, in application of the previous theorem. The pace of this book is very natural, the exercises adequate and well-selected. And in my experience, via this book students usually can handle the most some important topics and get a good structure feeling of this course. Highly recommended.

Rating:
Summary: Not enough for getting a complete perspective.
Review: 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).

Rating:
Summary: perhaps the best introduction to complex analysis
Review: 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).

Rating:
Summary: Excellent!
Review: This is a brief text on complex analysis aimed at the traditional junior-senior course. As a text it may be a little too succinct for the average undergraduate. For example, I have no intention of teaching out of it. However, its clarity and presentation is absolutely refreshing. I think it is one of the best books written on complex analysis in the last twenty years. I recommend this book to any student of complex analysis.

Rating:
Summary: Excellent!
Review: This is a brief text on complex analysis aimed at the traditional junior-senior course. As a text it may be a little too succinct for the average undergraduate. For example, I have no intention of teaching out of it. However, its clarity and presentation is absolutely refreshing. I think it is one of the best books written on complex analysis in the last twenty years. I recommend this book to any student of complex analysis.

Rating:
Summary: perhaps the best introduction to complex analysis
Review: 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.