The book under review is a second edition of a book by the same authors
and with the same title, also published by Springer in 2007.
It contains some amount of new information.
In particular it has an interesting section
with historical remarks by Ranjan Roy as an Appendix to
Chapter 1. More details on the new material will be given below.
As is stated in Chapter 1, Section 2, the main goal of the book
is to prove the equivalence of eight statements
all of them seminal in the theory of functions of one complex
variable, starting with the Cauchy-Riemann
equations and finishing with the Runge theorem on
approximation of complex analytic functions by
polynomials. Of course the key to obtain those
equivalences is the Cauchy integral theorem, which is
obtained in Section 4.6.
In Section 1.3 of Chapter 1 the plan of the proof
of these equivalences is explained. The proof
of (1)=>(2) is presented in Chapter 2 but the proof
of (8)=>(1) that closes the chain, has to wait until
Chapter 7, almost 200 pages later.
The authors use differential forms and Green's theorem
to obtain, locally, the existence of a primitive for a holomorphic
function on a domain D in the complex plane, which is crucial in
the theory. As is written in Section 4.8.
<>. This other approach is outlined
as Appendix II in Section 4.8 (not having appeared in the previous
edition of the book) and is developed in
most books on the topic. Let me quote just two:
For example, Conway and Rudin's books (see References 7
and 30 in the text).
Once the equivalence between the eight statements
already mentioned is obtained, a certain amount of important
classical results on holomorphic functions of one complex
variable is obtained. In particular, the Riemann
Mapping Theorem and the Dirichlet problem
on the existence of harmonic functions on a disc with
prescribed values on the boundary are study in depth.
In relation with the Dirichlet problem, a study of subharmonic
functions and the Green function is carried out (this is new in
this edition) and the Dirichlet problem is revisited using
subharmonic functions and Perron's Principle. Riemann
factorization theorem and infinite Blaschke products are also
Another topic, new in this edition, is the study of the
ring of holomorphic functions on a domain. It is worth
remarking Theorem 10.12 which characterizes the ring
isomorphism between two proper domains of the extended complex
In spite of the book covering almost all the classical results
of which are known as "Complex Variables", a proof of Picard's
theorem (mentioned in Section 6.2) will have been a good
closing for the book but, unfortunately, it is not included.
The book is carefully written and
each chapter has an interesting list of exercises.
I found it very useful as am Undergraduate and Graduate Text in Mathematics.