Supplementary Books to Complex Analysis of Rudin's RCA?

Question

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.

Answer 1

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!

Answer 2

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.