Supplements to Complex Analysis of Rudin-RCA?

[bacte2013]

Dear Physics Forum friends,

I will be doing a reading course in the complex analysis starting on this Fall Semester. The assigned book is Rudin's Real and Complex Analysis. From my understanding, Rudin treats complex analysis very elegantly, but very terse. I am curious if you could suggest some books in the complex analysis that can accommodate Rudin, with particular emphasis on the extensive treatment and/or clear explanations. I am embarrassed to ask my professor as I do not want to impose a bad impression on me.

Also, are previous chapters in Rudin-RCA a must requirement for later chapters in the complex analysis? I am currently reading through Berberian and Kolmogorov/Fomin to learn some basics of measure theory and banach space, but I have not completely learned them yet.

 

[micromass]

Simon's comprehensive course in analysis volume 2A and 2B
Freitag and Busam complex analysis
Needham visual complex analysis

 

[bacte2013]

Simon's comprehensive course in analysis volume 2A and 2B
Freitag and Busam complex analysis
Needham visual complex analysis

What are some strengths of Simons's two-volume books in the complex analysis? From the table of contents and preface, I got the impression that the book is very high-level and abstract. Also, does it assume the previous knowledge from the first volume (Basic Analysis)?

 

[The Bill]

What are some strengths of Simons's two-volume books in the complex analysis? From the table of contents and preface, I got the impression that the book is very high-level and abstract. Also, does it assume the previous knowledge from the first volume (Basic Analysis)?

Simon is a bit less brief and more detailed with some explanations, when you need that. Simon and Rudin complement each other rather well in this respect. If you're familiar with basic analysis, you should be fine with 2A and 2B. You may benefit from looking up the occasional term on MathWorld or Wikipedia, but that's true of almost any graduate text. And, you'll have Rudin to flip back and forth to too, so you're ahead of the game there.

 

[bacte2013]

Simon is a bit less brief and more detailed with some explanations, when you need that. Simon and Rudin complement each other rather well in this respect. If you're familiar with basic analysis, you should be fine with 2A and 2B. You may benefit from looking up the occasional term on MathWorld or Wikipedia, but that's true of almost any graduate text. And, you'll have Rudin to flip back and forth to too, so you're ahead of the game there.

Thank you for your answer. When you said "basic analysis", do you mean the introductory level (i.e. Rudin-PMA)? One thing that I am worried is that both Simon and Rudin have previous chapters in the real analysis (measure theory, etc.). I am currently learning it through Kolmogorov/Fomin, but I do not have good mastery of them yet.

Simons looks exciting. Perhaps I should tell my adviser if I can use Simon instead of Rudin as a main text. I also taken a look some books like Conway, Ahlfors, Freitag, and Needham, but they are not exciting as Rudin snd Simon.

 

[FactChecker]

I second @micromass 's recommendation of Needham's Visual Complex Analysis. It does not have rigorous proofs but it gives good intuitive motivations. I recommend it for filling in spots where more explanation is needed. The other books mentioned may be as good but I am not familiar with them.

 

[Ssnow]

I suggest also the classical '' Elementary Theory of Analytic Functions of One or Several Complex Variables'' of H.Cartan, Dover

 

[mathwonk]

I agree with Ssnow as Cartan is my absolute favorite complex book. I also like Lang's book, as well as George Mackey's. I do not like anything by Rudin, and find Ahlfors also not appealing for learning. In particular his treatment of Riemann surfaces (my specialty) is probably the least useful and least insightful possible. I like a lot of things about Hille's book. Fred Greenleaf's book is very easy to learn from. Here is a link to a series of answers to this question.

http://mathoverflow.net/questions/47732/specializing-in-complex-analysis