A good book on the complex analysis?

Question

I took complex analysis but I want to re-study it. When I took the class I studied the book written by stein( princeton series).

The book was quite good but the problems were too tough and I had no idea I was solving them in a right way.

I once read a intro. to the complex analysis by Silverman; the problems were not too hard but I think it lacks some rigor.

So anyone would recommend a very good book on the complex analysis? I would really appreciate it.

Answer 1

I think one classical book is Rudin's Real and Complex analysis.

The other book I used is "Complex Variables" from the Schaum's outlines collection, written by Murray R. Spiegel, Seymour Lipschutz, John J. Schiller and Dennis Spellman. This book contains a lot of fully solved exercices, really well explained.

Getting both these books, Rudin's for the theorical part and the other for the exercices could be a good choice.

Answer 2

If you are looking for an introduction to complex analysis with applications, such as the residue theorem or various contour integration techniques, a good choice would be

Complex Variables and applications by Churchill and Brown.

If you are looking for a more advanced introduction, a widely used book is

Complex Anlaysis: an introduction to the theory of analytic functions of one complex variable by Lars Ahlfors.

For a more historical background to the subject but, at a more rigorous level than either of the above texts (in the sense that there is a heavy use of analytic techniques and theorems acquired from real analysis) is

Analytic Function Theory Volumes I and II by Einar Hille.

Based on your situation, I would probably recommend Ahlfors strongly. I think Churchill and Brown would probably be too elementary. I don't think the problems in Hille's text are particularly difficult, and his work is wonderfully motivated but, I would say it is a matter of taste.

Edit: I would also add, Hille's text is written while some of the developers of the subject were still alive (i.e. Brouwer is mentioned as Brouwer 1881-). So his work is a little outdated with set notation but, the content is still very solid.

Answer 3

The introductory text I used was by Fundamentals of Complex Analysis by Saff and Snider. I really enjoy it. The exposition is conversational and lucid. The examples are well chosen... Clear enough to be solvable, but including enough of a trick or deviation to surprise or reward the reader. There are a good number of exercises with a range of difficulty at the end of each section but nothing crazy. The solution manuals are floating around for it online as well.

The authors choose to explore Cauchy's Theorem through both deformation of loops and again through vector calculus, so seeing the two approaches can be interesting. There are some quick forays that explore related topics, such as Julia and Mandelbrot sets or certain applications like the heat equation, but for the most part, the text sticks to teaching you the ropes.

Good text for self-learning the subject. As advertised, it is an introduction.