Complex Analysis: The Geometric Viewpoint (2nd ed.).
Godfrey, Stephen
Complex Analysis: The Geometric Viewpoint (2nd ed.)
Stephen G. Krantz Published by The Mathematical Association of
America, 2004) Hardcover, 219 pp., ISBN 08-8385-0354 US$51.95
Krantz has been a prolific writer of mathematics over the past 30
years with over 100 published papers and some 30 books to his name. The
book that is under review appears in the Carus Mathematical Monographs
series. This series is intended for people familiar with basic graduate
or advanced undergraduate mathematics who wish to extend their knowledge
without prolonged study of the mathematical texts and journals. This
books lives up to this pretext.
This is definitely not a book to give high school students unless
it is just to browse and see what some more advanced mathematics looks
like. The book would be ideal for the mathematically mature reader who
is interested in having an idea about what research is being undertaken
in this field of mathematics. I would recommend that the reader should
have completed a course in complex analysis and it would also be helpful
if the reader has been exposed to the concept of metric spaces.
In the main, the author does not divert from the chief topics but
at appropriate moments he does describe the historical development of
the ideas. For example, during the author's discussion of the
deeper meaning of the Schwarz lemma [A], we read that Ahlfors is said to
have commented that, "There is an almost trivial fact and anybody
who sees the need could prove it at once."
Fortunately for the reader, Krantz does prove this result.
The author begins with a review of functions of a complex variable,
also known as function theory, that will be used or extended later in
the book. The second chapter introduces the reader to some
"simple" concepts in differential geometry as well as the
Poincare metric.
The curvature of a metric is discussed in the third chapter and
then some invariant metrics are introduced. The book also discusses the
basics of Bergman theory, which is very readable as the author avoids
many of the details that arise by considering special spaces and not the
general Hilbert space theory.
In the last chapter the reader gets a glimpse of the theory of
several complex variables. This is an extremely difficult topic and even
today we are only just scratching the surface. For this reason Krantz
does not go into much detail and mainly discusses the Poincare theorem
that states there is no biholomorpic mapping of the bidisc to the ball
which is contrary to the case with a single complex variable.
Stephen Godfrey
Ref [A]: Ahlfors, L. (1938). An extension of Schwarz's lemma.
Trans. Amer. Math. Soc. 43, 359-364.
