# Supplementary Books to Complex Analysis of Rudin's RCA?

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 accomodate 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.

• You should ask your professor. Why would they get a bad impression? – Batman Aug 6 '16 at 3:32
• @Batman I do not want to leave an impression of not able to handle Rudin... – user205011 Aug 6 '16 at 3:56
• Rudin is very terse, but you should be more concerned about the knowledge than about your professor's opinion. I found Conway's "Functions of One Complex Variable" to be readable. – copper.hat Aug 6 '16 at 4:05
• @MathWanderer - I don't see why they would get that impression. It's completely acceptable (and normally provided by the instructor) to ask for a list of references of comparable / alternative presentations of required material. – Batman Aug 6 '16 at 12:55

Rudin is indeed very terse, but a useful reference. If you are looking for some geometric intuition on some of the fundamental results in complex analysis, I recommend Needham's "Visual Complex Analysis". His explanations are very well thought out, and the book sheds light on the subject in a way that normally you would not find in the classroom. It's considered a work of art by many.

In addition to Rudin and Needham, I referenced Complex Variables by Taylor up until about the analytic continuation section of the book (his explanations began to grow poor after that). Taylor includes some very helpful exercises and I would encourage you to try to complete the majority of them to gain a deeper understanding of the material.

Hope that helps!

• Thank you very much for your recommendation! Are insights and reasoning presented in Needham different from books like Rudin and Simon? From brief observation, it seems that Needham offers a lot of physical insight, but contains only few proofs to many theorems in the complex analysis.... – user205011 Aug 10 '16 at 20:53
• Anytime! Yes, I would say that the insights offered by Needham are quite different. The book is not sufficient by itself to complete an in depth course of self study, as (like you stated) he doesn't focus on proofs like Rudin / Simon. The physical intuitions offered by Needham in conjunction with Rudin / Simon is what I am recommending. A typical workflow I recommend would go as follows: (1) Read the statement of the theorem/concept (2) Read and understand *** Needham's insights (3) Use Needham's insights to help grasp proofs / explanations in Rudin/Simon – Anthony Felix Hernandez Jr. Aug 10 '16 at 22:33
• I think the overall difference is that authors like Rudin offer clever explanations / solutions to concepts that may otherwise not be intuitive. Needham gives you the intuition to understand how these clever approaches can work. – Anthony Felix Hernandez Jr. Aug 10 '16 at 22:37
• Thanks! My library has a copy of it, so I got it! As for Rudin, I might switch to Simon as he seems to cover very interesting materials. – user205011 Aug 11 '16 at 15:47

If you are looking for a lower-level text that introduces the basics of the material of complex analysis in a readable and pedagogical manner, I would suggest Stein and Shakarchi's Complex Analysis (Princeton Lectures in Analysis II). It also contains applications to Fourier analysis and number theory.

If you are looking for a less terse development of Rudin's chapters on harmonic functions and $H^p$ spaces, I would suggest the first four chapters of Koosis' Introduction to $H_p$ spaces (2nd edition) or the first two chapters of Garnett's Bounded Analytic Functions.

• Thanks for the suggestion! I actually decide to read Simon's book for the reading course. I had taken a look at Stein/Shakarchi's book, but I did not like it. Perhaps Koosis and Garnett will work great with Simon too? – user205011 Aug 13 '16 at 0:15