##
**Functions of one complex variable.**
*(English)*
Zbl 0277.30001

Graduate Texts in Mathematics. 11. New York-Heidelberg-Berlin: Springer-Verlag. xi, 313 p. DM 41.10; $ 16.90 (1973).

This is a good textbook written on a first course in the theory of functions of one complex variable. The definitions and theorems etc. are stated clearly and precisely. Most of the proofs are presented in detail and when this is not the case the author has clearly stated what is missing and asked the reader to fill in the gaps.

Contents: Chapter 1 studies the field of complex numbers and the geometric representation of complex numbers. Chapter 2 is on metric spaces and the topology of complex numbers. In Chapter 3 are given some elementary properties and examples of analytic functions. Chapter 4 initiates the student to complex integration. Chapter 5 is devoted to singularities and the argument principle. In Chapter 6 is given the maximum modulus theorem, Schwarz’s lemma, convex functions and Hadamard’s three circle theorem. Chapter 7 studies compactness and convergence in the space of analytic functions. Chapter 8 includes Runge’s theorem and its application to obtain a more general form of Cauchy’s theorem. In Chapter 9 analytic continuation, analytic manifolds and covering surfaces are discussed. Chapter 10 contains harmonic functions, subharmonic and superharmonic functions, the Dirichlet problem and Green’s functions. The last two Chapters 11 and 12 are devoted to the study of entire functions and Picard’s theorems. With regard to Picard’s theorem the author states that the proof given here is based on elementary arguments whereas the proof in most other books uses the modular function.

[The reviewer notices that the proof of Picard’s theorem given here is similar to the one given in “Lectures on the theory of functions of a complex variable. I: Holomorphic functions” (1960; Zbl 0093.26803) by G. Sansone and J. Gerretsen.]

The reviewer finds that perhaps due to poor proof reading a number of minor errors have crept into the book (see pages 9, 23, 34, 77, 79, 105, 124, 166, 259, 267). However, the subject matter has been presented in simple clear, precise language and the book should prove useful to the students taking up first course in theory of functions of one complex variable.

Contents: Chapter 1 studies the field of complex numbers and the geometric representation of complex numbers. Chapter 2 is on metric spaces and the topology of complex numbers. In Chapter 3 are given some elementary properties and examples of analytic functions. Chapter 4 initiates the student to complex integration. Chapter 5 is devoted to singularities and the argument principle. In Chapter 6 is given the maximum modulus theorem, Schwarz’s lemma, convex functions and Hadamard’s three circle theorem. Chapter 7 studies compactness and convergence in the space of analytic functions. Chapter 8 includes Runge’s theorem and its application to obtain a more general form of Cauchy’s theorem. In Chapter 9 analytic continuation, analytic manifolds and covering surfaces are discussed. Chapter 10 contains harmonic functions, subharmonic and superharmonic functions, the Dirichlet problem and Green’s functions. The last two Chapters 11 and 12 are devoted to the study of entire functions and Picard’s theorems. With regard to Picard’s theorem the author states that the proof given here is based on elementary arguments whereas the proof in most other books uses the modular function.

[The reviewer notices that the proof of Picard’s theorem given here is similar to the one given in “Lectures on the theory of functions of a complex variable. I: Holomorphic functions” (1960; Zbl 0093.26803) by G. Sansone and J. Gerretsen.]

The reviewer finds that perhaps due to poor proof reading a number of minor errors have crept into the book (see pages 9, 23, 34, 77, 79, 105, 124, 166, 259, 267). However, the subject matter has been presented in simple clear, precise language and the book should prove useful to the students taking up first course in theory of functions of one complex variable.

Reviewer: Ram Murti Goel (Patiala)

### MSC:

30-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable |

31-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to potential theory |