Specializing in Complex Analysis [closed]

Question

May someone kindly provide a useful list of books on complex analysis that would be appropriate for a graduate student intending to specialize in that area.

Answer 1

In addition to the beginning complex books mentioned above, I suggest a few others, mostly older sources:

1. my all time favorite introduction is the book by Henri Cartan. It is extremely clear and elegant, and includes a brief introduction to the several variable case, and Riemann surfaces. In contrast to Ahlfors, who bases the treatment on integration, Cartan begins with a very nice treatment of formal and convergent power series. The Riemann mapping theorem is proved very clearly.

2. one of the most comprehensive introductory books is the 2 volume treatise by Einar Hille. It is one of the few which proves the big Picard theorem as well as the little one. It is also inexpensive (a Chelsea book). It includes a proof of the polygonal Jordan curve theorem.

3. Lang's complex analysis is, as are all of his books, well written and focuses on important aspects and complete proofs of every topic chosen. The explanations are concise but insightful and clear. Like Cartan, he begins with power series. It is also relatively cheap.

4. The out of print book by Frederick Greenleaf is one of the most thorough and carefully written, intended for the beginner undergraduate. It makes a nice more detailed complement to the deeper more difficult Lang.

5. A masterful and incredibly concise book, with an excellent array of problems with solutions is the 4-5 volume set of small paperbacks by Konrad Knopp. Also cheap Dovers.

6. The lectures by George Mackey, Lectures on the theory of functions of a complex variable, Van Nostrand Mathematical Studies, https://www.amazon.com/Lectures-Functions-Complex-Variable-Mathematics/dp/0442050682 are an amazing testimony to the scholarly erudition of this author. This book is very theoretical, and essentially flawless, even though it is simply his lecture notes for the course, written down for the class's benefit at the time. Every topic is treated in the most precise and excellent way. Analytic continuation is well treated, and the theory of abelian integrals is outlined at the end through Riemann Roch. Unfortunately these final proofs are not included. The book is a little hard to find.

7. Many people recommend Ahlfors, including professional analysts, and it does have a very nice treatment of infinite products. I found it overly concise in places treating analysis concepts such as the J function, and overly verbose in others concerning elementary plane topology. Several proofs such as the integral expansion of an analytic function seem less clear to me than the direct expansion of the integral kernel found elsewhere. The discussion of an algebraic function is also relatively unenlightening to me concerning the geometry of the subject. After several encounters, it does not seem to me the ideal place to learn the subject, but it seems prudent to eventually master his discussion since professional analysts like it.

8. The book of Jones and Singerman gives a good treatment to the geometric side of complex analysis.

9. After learning the basic one variable material in the plane, an excellent source for the extension to the case of Riemann surfaces is the Princeton lecture notes by Robert Gunning. It has a very good introduction to sheaf cohomology including a nice proof of Serre duality with complete discussion of the relevant functional analysis made easier by placing a Hilbert space structure on the relevant cocycle spaces. If you cannot find it, Gunning has a free updated version, but more concise, and incomplete, on his website.

Great classic works on Riemann surfaces include Riemann*, Hermann Weyl, and the 3 volume work by C.L.Siegel.

10. -11) For the extension to the several variables case there are two classics, the book of Gunning and Rossi, and the book of Hormander. Gunning and Rossi gives a thorough treatment of the Oka theory of sheaf cohomology and a lot of useful material, but less clearly written, on analytic varieties. Hormander is very terse, covering all of one complex variable in a chapter one that is about 20 pages long. Chapter 2 includes the fundamental theorem of Hartogs that functions analytic on a punctured nbhd of (0,0) extend across the origin. The development is based on the solution of the C - infinity "dbar" problem.

Other books that look good but I have not read are those of Forster on Riemann surfaces, and of Krantz on several complex variables, and of Ebeling on some geometry of Riemann surfaces, several complex variables and singularities. Joe Taylor also has an interesting book.

A useful book on complex manifolds is that of Kodaira and Morrow, and the classic on deformations of complex manifolds is that of Kodaira. Andre Weil's classic Varie'te's Kahleriennes is also recommended.

I am not a complex analyst, but a complex algebraic geometer, so my recommendations should be taken in that light.

Answer 2

Since complex analysis is so broad, I'll mention three I like:

Visual Complex Analysis by Tristan Needham, Oxford Press has a unique geometric approach.

Princeton Lectures in Analysis II, Complex Analysis, by Stein and Shakarchi, PUP, is beautifully written.

Complex Variables by Flanigan is an inexpensive Dover which helpfully reviews vector calculus in the plane and harmonic functions extensively. Once this is done the complex theorems come easily.

Answer 3

I love Sarason's book as an introduction. It was originally (I think) published by Henry Helson in his garage. It's just beautifully clean and goes as far as a proof of the Riemann mapping theorem.

It's recently been taken over by the AMS.