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

studied.

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

plane.

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.