Complex analysis.

*(English)*Zbl 0978.30001
Undergraduate Texts in Mathematics. New York, NY: Springer. xvi, 478 p. (2001).

The author of this fine textbook is a prominent function theorist. He leads the reader in a careful but far-sighted way from the elements of complex analysis to advanced topics in function theory and at some points to topics of modern research. Geometric aspects (the Riemannian point of view) and purely analytic aspects (the Weierstrass point of view) are treated in a balanced way, figures appear frequently in the text.

Part I of the book (7 chapters, 223 pages) contains basic material up to Laurent series and the residue calculus. Functions like \(\sqrt z\), \(\exp(z)\), \(\sin(z)\), \(\sqrt{z(1-z)}\) including a ‘light’ discussion of their Riemann surfaces appear already in Chapter I. Line integrals and complex integration are treated via their real-analytic counterparts. In particular, Green’s theorems are used freely. There are also sections on fluid dynamics and other applications. Doubly periodic functions and complex Fourier series are part of Chapter VI, and in Chapter VII Cauchy principal value integrals are treated besides the standard topics of the residue calculus.

Part II (4 chapters, 90 pages) begins with the argument principle and Rouché’s theorem, followed by the ‘jump theorem’ and a thorough discussion of simple connectivity. The Schwarz lemma, the Schwarz reflection principle, and the Schwarz-Christoffel formula follow in Chapters IX, X and XI. Included in Chapter IX are the basic notions of hyperbolic geometry in the unit disk. Compact families are introduced with the Riemann mapping theorem in view, which is proved at the end of Part II.

Part III (5 chapters, 131 pages) is very rich in advanced topics. Chapter XII begins with a thorough treatment of compact families of meromorphic functions. The key theorem is Marty’s theorem which requires the uniform boundedness of the spherical derivatives on compact subsets. Zalcman’s lemma and Montel’s theorem (3 values omitted) give criteria for normality. Picard’s theorems (big and little) follow. The remainder of Chapter XII (3 sections) gives an ab ovo introduction into modern iteration theory of rational functions. Fatou sets, Julia sets, basin of attraction, periodic points are some of the topics. In particular, one is interested in the connectedness of the Julia set. The iteration of quadratic polynomials \(P(z)= z^2+ c\) leads to the Mandelbrot set \(M\). Runge’s approximation theorem and some of its applications are in Chapter XIII, followed by the theorem of Mittag-Leffler and Weierstrass. The highlight of Chapter XIV is the prime number theorem proved via a Tauberian theorem for Laplace integrals. Dirichlet series come along the way. In Chapter XV the general Dirichlet problem is discussed and solved via the Perron method, and Riemann’s mapping theorem is proved once more. Also treated are Green’s function and harmonic measure.

In the last chapter a careful introduction to Riemann surfaces is given, the Dirichlet problem and the general Green’s function are studied as well as the bipolar Green’s function. All this culminates in Koebe’s uniformization theorem: Every simply connected Riemann surface is conformally equivalent to either one of \(\mathbb{C}\), \(\mathbb{C}^*\) or \(\mathbb{D}\).

At the beginning of the book there are two pages of historical remarks, just enough for a text on complex analysis. We found, however, that a special feature of the book is the wealth of exercises at the end of each section, ranging from easy and straightforward ones to demanding ones which sometimes prepare the ground for later developments. Hints for solutions are given for many of the exercises.

Altogether, the author has given us a wonderful textbook for use in classroom and in seminars.

Part I of the book (7 chapters, 223 pages) contains basic material up to Laurent series and the residue calculus. Functions like \(\sqrt z\), \(\exp(z)\), \(\sin(z)\), \(\sqrt{z(1-z)}\) including a ‘light’ discussion of their Riemann surfaces appear already in Chapter I. Line integrals and complex integration are treated via their real-analytic counterparts. In particular, Green’s theorems are used freely. There are also sections on fluid dynamics and other applications. Doubly periodic functions and complex Fourier series are part of Chapter VI, and in Chapter VII Cauchy principal value integrals are treated besides the standard topics of the residue calculus.

Part II (4 chapters, 90 pages) begins with the argument principle and Rouché’s theorem, followed by the ‘jump theorem’ and a thorough discussion of simple connectivity. The Schwarz lemma, the Schwarz reflection principle, and the Schwarz-Christoffel formula follow in Chapters IX, X and XI. Included in Chapter IX are the basic notions of hyperbolic geometry in the unit disk. Compact families are introduced with the Riemann mapping theorem in view, which is proved at the end of Part II.

Part III (5 chapters, 131 pages) is very rich in advanced topics. Chapter XII begins with a thorough treatment of compact families of meromorphic functions. The key theorem is Marty’s theorem which requires the uniform boundedness of the spherical derivatives on compact subsets. Zalcman’s lemma and Montel’s theorem (3 values omitted) give criteria for normality. Picard’s theorems (big and little) follow. The remainder of Chapter XII (3 sections) gives an ab ovo introduction into modern iteration theory of rational functions. Fatou sets, Julia sets, basin of attraction, periodic points are some of the topics. In particular, one is interested in the connectedness of the Julia set. The iteration of quadratic polynomials \(P(z)= z^2+ c\) leads to the Mandelbrot set \(M\). Runge’s approximation theorem and some of its applications are in Chapter XIII, followed by the theorem of Mittag-Leffler and Weierstrass. The highlight of Chapter XIV is the prime number theorem proved via a Tauberian theorem for Laplace integrals. Dirichlet series come along the way. In Chapter XV the general Dirichlet problem is discussed and solved via the Perron method, and Riemann’s mapping theorem is proved once more. Also treated are Green’s function and harmonic measure.

In the last chapter a careful introduction to Riemann surfaces is given, the Dirichlet problem and the general Green’s function are studied as well as the bipolar Green’s function. All this culminates in Koebe’s uniformization theorem: Every simply connected Riemann surface is conformally equivalent to either one of \(\mathbb{C}\), \(\mathbb{C}^*\) or \(\mathbb{D}\).

At the beginning of the book there are two pages of historical remarks, just enough for a text on complex analysis. We found, however, that a special feature of the book is the wealth of exercises at the end of each section, ranging from easy and straightforward ones to demanding ones which sometimes prepare the ground for later developments. Hints for solutions are given for many of the exercises.

Altogether, the author has given us a wonderful textbook for use in classroom and in seminars.

Reviewer: Dieter Gaier (Gießen)

##### MSC:

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