Math 503: Complex Analysis, in the Fall, 1997, semester
Preface Background Syllabus Homework Links to 503 technical pages The text

## The textbook and some alternatives

The text I've chosen is An Introduction to Complex Function Theory, by B. Palka, Springer-Verlag (1991). I've taught Math 503 a number of times. I've used different texts each of the preceding three occasions (texts written by Ahlfors, Conway, and Remmert respectively). I requested information from graduate students contrasting these texts and asking about any others. My stated criteria were:

• A useful lifetime addition to a personal technical library, having an interesting systematic exposition of the area.
• Easy enough to read, using mostly standard notation and presenting most of the standard results, with enormous effort not required.
• Helpful in preparing for our qualifying exams.

I think that my choice fulfills much of this list. The last time I presented this course I wrote the following:

There are hundreds of books on this subject. A few of them are listed below. They are the first texts I tend to look at when I want to learn some complex variables fact. You may develop different preferences. You can have hours of fun by going to the library and investigating the region around QA331 (as I did when I was a graduate student). I recommend that you do this.

Here is a short list of a few other complex variables texts, including some of my personal favorites:

• Complex analysis: an introduction to the theory of analytic functions of one complex variable, by L. Ahlfors, McGraw-Hill (1979).
This is the latest edition of an awesomely correct and insightful book. Beginners and more experienced readers sometimes find the insights difficult to grasp, however.
• Functions of one complex variable, by J. Conway, Springer-Verlag (1986).
This is a pleasant and efficient book. It does not seem to me to have a strong point of view, but it presents the material in a straightforward fashion.
Two classical references:
• Analytic Functions, by S. Saks & A. Zygmund, Elsevier (1971).
This text was first published more than 50 years ago. It is clear, clever, and correct, with many wonderful problems and some ways of approaching the subject that are different from those in many other books.
• Analytic Function Theory, by E. Hille, Chelsea (1973).
This is the most recent reprint of a two volume work by my professional great-great-grandfather. It has many, many facts: maybe more than you will ever want to know about certain parts of complex analysis.
Some books written fairly recently by other masters of complex analysis:
• Introduction to Complex Analysis, by K. Kodaira, Cambridge University Press (1984).
This has an extremely careful discussion of some of the point-set topology of the plane, and includes a number of very graceful proofs.
• Complex analysis in one variable, by R. Narasimhan, Birkhauser (1985).
This is a remarkably condensed presentation of many results of complex analysis, including some proved only in recent years. Lots of hard analysis, especially illustrating connections with partial differential equations.
• The Theory of Complex Functions, by R. Remmert, Springer-Verlag (1991).
I used this text the last time I taught the course. Remmert carefully and leisurely explores different approaches to the subject and discusses some of the human aspects, including a few of the historical dead ends. I find his presentation interesting, friendly, and thorough. Student reaction was not uniformly positive. The text does not contain certain results that we often hope to cover in 503 but rarely do: the Mittag-Leffler Theorem, the Weierstrass Factorization Theorem, and the Riemann Mapping Theorem.
A standard undergraduate text:
• Complex Variables and Applications, by R. Churchill and J. Brown, McGraw-Hill (1990).
This is the latest of numerous editions. This book has many elementary examples and presents simple versions of the powerful foundational theorems of complex analysis. It also discusses some interesting applications to problems in physics and engineering.
I investigated a number of books published recently, and looked at books recommended by graduate students. I chose the text by Palka. It is certainly appropriate for a graduate level course in spite of its inclusion in the Springer series of Undergraduate (!) Texts in Mathematics. You can examine its contents . Palka's text has a number of neatly worked out examples and covers the material appropriately. There's a good selection of exercises. I've only found one that's wrong so far: either careful editing or lazy reading! Perhaps the only deficiency I've noticed so far is that there's no systematic exposition of the D-bar equation which is a strength of Narasimhan's text.

Note, please, that the cost of Palka's book is approximately half the cost of Ahlfors's book.

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