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
- 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,
This is the latest edition of an awesomely correct and
insightful book. Beginners and more
experienced readers sometimes find the insights difficult to grasp,
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
Two classical references:
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
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,
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
The Theory of Complex Functions, by R. Remmert,
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:
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.
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.
Note, please, that the cost of Palka's book is approximately half the
cost of Ahlfors's book.
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