MAT 528 -- Complex Variables II 
Additional Reading

References:

• Reinhold Remmert, Theory of Complex Functions, Springer-Verlag Graduate Texts in Mathematics #122, New York, 1991.

  This book is a translation of the second edition of Funktionentheorie I, Grundwissen Mathematik 5, Springer-Verlag, 1989 by the same author. It's worth reading just for the copious historical notes, but the approach to the classical transcendental functions, especially Eisenstein's theory of the trigonometric functions, is also quite appealing.

• John B. Conway, Functions of One Complex Variable I, (2nd ed.) Springer-Verlag Graduate Texts in Mathematics #11, New York, 1978.

  Conway's book is widely used; I've taught MAT 528 from it previously myself. Although Conway omits elliptic functions, he covers the Picard theorems in more detail than Lang, and also covers Runge's theorem, which Lang omits. Conway's approach to contour integration is one of the most general I've seen: the use of Riemann-Stieljes integrals makes his development somewhat more intricate, but with a pay-off of weaker hypotheses for many of his results.

• D. H. Luecking and Lee A. Rubel, Complex Analysis: A Functional Analysis Approach, Springer-Verlag, New York, 1984.

  As most of us who studied complex analysis after 1980 or so are aware, there is a nice proof of Runge's theorem due to Sandy Grabiner that uses ideas from Banach algebras to avoid pole pushing. Rubel and Luecking's book extends this approach to the whole subject. The late Lee Rubel was an expert on entire functions, and attending his lectures was always an inspiration. This book would make a good reference for a further course in complex analysis or a seminar series.

• Joseph Bak and Donald J. Newman, Complex Analysis, (2nd ed.) Springer-Verlag, New York, 1997.

  Bak and Newman's text is actually undergraduate level; nevertheless there are sufficiently many charming examples and applications discussed here that makes it definitely worth a look. See especially the section on applying contour integration to estimating sums, an entire function bounded in every direction, and the partition problem in chapter 19.

• John Stalker, Complex Analysis: Fundamentals of the Classical Theory of Functions, Birkhäuser, Boston, 1998.

  Stalker's book uses the classical special functions to motivate his development. This might be a good book for a topics course.

• E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, London, 1969.

  Despite the title, this book is anything but modern. Nevertheless, as noted by Stalker, modern introductions to analysis tend to follow the pattern as laid down by Whittaker and Watson. In part I, the foundational material is given, with basic definitions and theorems. In part II, we find the examples and applications. Despite being somewhat old-fashioned, there are a wealth of interesting, concrete "Tripos style" problems involving special series and integrals that will entertain and delight.

• Stanislaw Saks and Antoni Zygmund, Analytic Functions, (2nd ed.) Polska Akademia Nauk, Warsaw, 1965.

  This is the text we used when I learned the subject. Unfortunately, it is now out of print.

• Lars V. Ahlfors, Complex Analysis, McGraw-Hill, New York, 1966.

  This was another popular text in years gone by. It was once said that learning complex analysis was like learning to walk. Before one could walk, one must learn to crawl on "ahlfors".

• C. Carathéodory, Theory of Functions of a Complex Variable, (F. Steinhardt, trans.) Vol. 1, Chelsea Publishing, New York, 1983.

  A classic text by one of the masters.

• Einar Hille, Analytic Function Theory, (2 Volumes), Blaisdell Publishing Company, New York, 1963.

  Another classic.

• Konrad Knopp, Theory of Functions, (5 Volumes), Dover, New York, 1975.

  Knopp's books have many solved problems worked out in detail. As a graduate student, I found it helpful in building my confidence to work on the problems in Knopp and then compare my solutions with his. Two further books with good problems include:

• Ruel V. Churchill and James Ward Brown, Complex Variables and Applications, McGraw-Hill, New York, 1984,

  and

• Jerrold E. Marsden and Michael J. Hoffman, Basic Complex Analysis, (2nd ed.) W. H. Freeman and Company, New York, 1987.

  No list would be complete without the inclusion of

• George Pólya and Gabor Szegö, Problems and Theorems in Analysis, (Two Vols.) Springer-Verlag, Berlin, 1998.

  Said to be the inspiration for generations of research mathematicians, this classic two-volume set consists of problems from all branches of analysis. The problems are artfully constructed and arranged so as to coax solvers gradually into active research.