Information for Math 503, fall 2007
About the course
Office: Hill 542; (732) 445-3074;
Office hours: Tuesday 2PM--5PM, and by appointment. Please feel free
to visit and talk. I also hope to be at "teatime" frequently.
I likely will do all of the grading, unless I get tired! I'll also do
the lecturing, although maybe I can get some members of the class to
My own work has been mostly in complex analysis and partial
differential equations. I have also worked a bit in discrete math, and
I am interested in aspects of math education, including outreach,
efficiency, and the sustainability of instructional improvements. I
have done some academic administration: I've been in charge of the
undergraduate program, and have been Graduate Director several times.
Office: Hill 606; (732) 445-6540
Office hours: Thursday 4:00 to 6:00 PM (only for Math 503!) and by
This year for the first time we'll have T.A.'s in the basic graduate
courses. Their roles are still not precisely defined, but Mr. Osorio
will at least have office hours specifically for Math 503 so you can
ask him questions (although I rarely bite). His qualifications for
helping in 503 include his undoubted expertise in analysis, and
his survival of Math 503 when I taught the course a few years ago.
Mr. Osorio is a doctoral student in mathematics, working with Professor Paul Feehan on
The text and other references
I chose the book Function Theory of One Complex Variable by
Robert Greene and Stephen Krantz. It is useful to have a specific
text for this course so that everyone can refer to it (at least
initially). I don't guarantee to always follow the approach in the
book, though! The book is in its third edition which is a positive
indication to me. This is a contemporary
book for graduate complex analysis first published about 10 years
ago. For it to appear in three editions means that people must be
buying it (more "customers"!), and that the errors and
infelicities originally present must have been decreased. Also the
price is not highly objectionable (compare it to the list price for
Ahlfors' text, for example).
Greene and Krantz are accomplished researchers and expositors in
complex analysis. I have read nice papers by both of them. Their book
has useful review material, and the discussions seem accessible. The
problems vary in difficulty, and some seem to be quite elementary.
Today (9/2/2007) I asked the Mathematical Sciences Library to put its
copy of the third edition on reserve. This means that you can go to
the desk and request the library copy and use it (in the library) for
two hours. This may be convenient if you need to refer to the book and
don't happen to have a copy with you.
Google just gave me more than 1.5
million links for the phrase "complex analysis". Here I will discuss
Most research libraries in the U.S. classify books with call
numbers using the Library of Congress system. The mathematics
classification is QA, and,
inside QA, texts concerning the material of this course are usually in
"theory of functions" (that's QA 331 to QA 355). For example, one
edition of the classical text of Ahlfors entitled Complex
analysis; an introduction to the theory of analytic functions of one
complex variable has call number QA331.A45 1966.
You can spend some happy hours as I did when I was a graduate student,
and look at the shelves around QA331. You will likely discover some
interesting objects. As I will mention in class, 80 to 85% of the
subject matter of Math 503 could have been discussed in 1890. But the
material is so attractive that people continue to discover "new" ways
to discuss it. The Rutgers
libraries can be searched on line, but looking at the shelves can
be interesting and sometimes serendipitous.
I will likely refer to some of the following books while preparing
lectures in this course. I own copies of many of them (bought with my
own bucks!) and have enjoyed contrasting the styles and examples of
the authors. My remarks are my own brief "appreciations" of these
texts and are not meant to be complete or authoritative. I note which
have been used in Math 503 as principal texts (by others and by me).
- Complex analysis; an introduction to the theory of analytic
functions of one complex variable by Lars V. Ahlfors
The author was a great Finnish guru (here I am using "guru" to mean
"an influential teacher", not a spiritual leader!). Although I learned
from this in the first complex analysis course I took, and I used it
as a text when I first taught the course, I've decided that it is just
too inaccessible. A fond and irritating memory when I was learning the
subject goes like this: I spent two hours (really!) reading and
rereading a short paragraph, trying to understand it. I finally did,
and then decided that this was the best and only way to phrase the
explanation. But such enlightenment comes sometimes with great effort.
QA331.A45 1966 Used as a text in Math 503.
- Complex variables: an introduction by Carlos A. Berenstein and
A large portion of this thick and thorough book is devoted to carrying
out the novel approach sketched in the first chapter of
Hörmander's book, mentioned below. The book can introduce you to
many areas of current research. QA331.7.B46 1991
- Theorie der analytischen Funktionen einer komplexen
Veränderlichen by H. Behnke und Friedrich Sommer.
An authoritative German text. QA331.B348 1962
- Elementary theory of analytic functions of
one or several complex variables by Henri Cartan.
The text begins by defining a holomorphic function as one which is
locally the sum of a power series, and then recovers other equivalent
statements later. The approach is more algebraic than most texts at
this level. QA331.C3213 Used as a text in Math
- Introduction to complex variables
and applications by R. V. Churchill.
U.S. undergraduate text. QA331.C524
- Functions of a complex variable : theory and technique by
Carrier, Krook, and Pearson.
A text used for for undergraduate and
graduate courses, aimed more towards engineering
- Functions of one complex variable by John B. Conway.
A standard U.S. text for the first semester graduate course in complex
variables: good, straightforward exposition and problems. QA331.C659
Used as a text in Math 503.
- Complex variables Stephen D. Fisher.
Currently used in our undergraduate course. Has been reprinted by
Dover in a cheap edition. QA331.F589 1986
- Applied and computational complex analysis by Peter
A formidable three-volume work discussing the topics of the title in
detail. QA331.H453 1974
- Analytic function theory by Einar Hille.
Two volumes of classical complex analysis by my professional
great-grandfather (so it must be good!). QA331.H54 1973
- An introduction to complex analysis in several variables by
Although almost 30 years old, the first chapter of this book is a
startling view of one variable complex analysis by one of the
originators of modern partial differential equations. Its view of the
subject changed everything! The first chapter gives an approach quite
distinct from the traditional 1890's style. QA331.H64
- Le calcul des résidus et ses applications à la
théorie des fonctions by Ernst Lindelöf.
The early text discussing how to compute with residues (hey, it's all
wired into Maple now, right?
- Theory of functions, by K. Knopp.
A multivolume work, available in a cheap Dover edition. QA331.K713
- Introduction to complex analysis by Kunihiko Kodaira.
An introductory text by a great Japanese complex analyst, indeed, a
Japanese guru. The book is full of insight and perhaps deceptive
simplicity. For example, if I can once again master it, I will try to
give Kodaira's wonderfully arranged proof of the differentiability of
a power series inside its radius of convergence. QA331.K718 1984
- Complex analysis by Serge Lang.
An introductory complex analysis text by a prolific and informed
author. QA331.L255 1985 Used as a text in Math
- Complex variables by Norman Levinson and Raymond
A text on the undergraduate/graduate (intermediate) level. QA331.L47
- The theory of analytic functions by A.I. Markushevich.
The standard Russian text for a generation. QA331.M363 1983
- Basic complex analysis by Jerrold E. Marsden.
An "undergraduate" (nearly!) text by an accomplished expositor and
applied mathematician. QA331.M378
- Complex analysis in one variable (second edition) by
Raghavan Narasimhan and Yves Nievergelt.
An excellent book for analysts by Narasimhan, who is a wizard complex
analyst and also a great expositor. Many avenues to further advanced
study are given here. A substantial problem section by Nievergelt has
been added to the second edition. QA331.N27 2000 Used as a text in Math 503.
- Introduction to complex analysis by Rolf Nevanlinna and
There's a subject called "Nevanlinna theory" which explores the
interaction between geometry and complex analysis. This text,
coauthored by R. Nevanlinna, has some interesting geometric
insights. QA331.N4713 1982 Used as a text in Math
- An introduction to complex function theory by Bruce P. Palka.
A well-written book in the Springer undergraduate text series. It fits
more nearly the standard first U.S. graduate course in complex
analysis (such as our Math 503). QA331.7.P35 1991 Used as a text in Math 503.
- Theory of complex functions by Reinhold Remmert.
Students worry because invention of lovely and intricate mathematics
seems nearly impossible. Please look at this book. Remmert
investigates the history of some obvious definitions and theorems. He
shows how the "ancient masters" frequently made the wrong definitions
(at least, tried to work with inconvenient definitions) and sometimes
gave defective proofs of both false and true theorems. The
intertwining of history and exposition is wonderfully done, and there
are some very interesting problems. QA331.R4613 1991 Used as a text in Math 503.
- Real and complex analysis by Walter Rudin.
Of course this contains most of the shortest and cleverest proofs. It
can sometimes be frightening to read, because no human being could
ever invent all these clever ideas. Much introductory real variables and
functional analysis is also discussed in this book.
QA300.R82 1987 (NOT 331!)
- Analytic functions by S. Saks and A. Zygmund.
A wonderful book, with an amazingly elegant and "modern" exposition of
the subject (the original edition is 80 years old!). There are some
wonderful and difficult problems (the third problem on problem set 0
comes from this book). QA331.S315 1971
- Complex Analysis: Fundamentals of the Classical Theory of
Functions by John Stalker.
A special function approach to complex analysis, studying specific
properties of important transcendental functions. Students who may be
interested in mathematical physics and number theory might find this
text very appealing. The library does not seem to own a copy.
- Complex analysis by Elias M. Stein and Rami Shakarchi.
Eli Stein, an analyst with wide and deep interests, is still
flourishing down the road (at Princeton). He has coauthored a lovely
book for the motivated reader. QA331.S74 2003 Used
as a text in Math 503.
- The theory of functions by E. C. Titchmarsh.
One of the classic English textbooks. QA331.T5 1969
- Visual complex analysis by Tristan Needham.
A modern text by an author who believes in the power of pictures. If
you think visually, you should really look at this book. QA331.7.N44
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