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Complex Analysis: An Invitation

Murali Rao, Henrik Stetkær, Søren Fournais, and Jacob Schach Møller
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Tushar Das
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This refreshing graduate text on function theory in one complex variable extends a previous edition (World Scientific, 1991) by Murali Rao and Henrik Stetkær. The earlier edition consisted of fourteen chapters which remain to make up the core of the present incarnation. They contain more than enough material for a successful year long course. For the second edition, the authors have joined forces with Søren Fournais and Jacob Schach Møller who have written six new chapters to follow Rao and Stetkær’s core.

A word about prerequisites. It is the authors’ “impression that courses in complex function theory these days at many places are postponed to leave space for other topics like point set topology and measure theory.” They therefore assume a background in point set topology and real analysis (both single and multivariable). For instance, this includes a liberal use of Lebesgue’s dominated and monotone convergence theorems starting from a third of the way into the book. The authors have used this text in both undergraduate and graduate courses at Aarhus University. Perhaps a strong class of highly motivated seniors would be able to keep up with the book. Given the right audience, the book could also be profitably used in a seminar, or to guide an independent study.

A particularly nice feature of this textbook is that throughout it, and especially in the first fourteen chapters, the authors have sprinkled citations of expository pieces taken from the Monthly as well as from Mathematics Magazine, the Intelligencer, and even the Proceedings of the AMS. This is a great source for both students as well as their teachers. In fact, many of their proofs come from such alternative sources. The book also is an excellent source of problems, close to 300 in total. There are a few very minor typos, all of which are easily spotted, providing brief amusement and then easily corrected. The notation used is mostly standard, except for the few matrices that came with (curly) braces!

The book begins with power series, Abel’s theorem and its partial inverse due to Tauber. On page 8 we are presented with a general version of Liouville’s theorem: a power series that grows at most polynomially at infinity is a polynomial. This is followed by a standard complex analysis core: basic complex calculus (complex differentiability and line integrals), the exponential function, logarithms and the winding number, the Cauchy-Goursat integral theorem and consequences, a proof of the open mapping theorem avoiding topology, and isolated singularities and Cauchy’s residue theorem. This takes up the first six chapters.

The next two chapters (7 and 8) take up Picard’s theorems and the Riemann mapping theorem, respectively. The standard presentation of Picard’s theorems is followed by a more geometric treatment (via the curvature of hyperbolic metrics) that is probably absent in most basic textbooks. The authors follow the paper by David Minda and Glenn Schober, “Another elementary approach to the theorems of Landau, Montel, Picard and Schottky,” (Complex Variables Theory Appl. 2 (1983), no. 2, 157–164).

For a text at this level, it would be reasonable for the Riemann mapping theorem to be followed by the uniformization theorem, see e.g. Fisher-Hubbard-Wittner’s neat approach in “A proof of the uniformization theorem for arbitrary plane domains,” (Proc. Amer. Math. Soc. 104 (1988), no. 2, 413–418). If a review must contain a quibble, however, here is mine: there is not a single mention of Riemann surfaces throughout the text. The inclusion of such a rich thread that ties together analysis, algebra and topology would certainly have made this already rich book even richer!

Chapters 9 and 10 present the basics of meromorphic functions. Highlights include Runge’s theorem and the representation theorems of Weierstrass and Mittag-Leffler. Chapter 11 on the prime number theorem provides an example of the authors’ succumbing “to the temptation of making a digression to an interesting topic”.

This is followed by Chapters 12 and 13 on harmonic and subharmonic functions. Here we return to the theme of Liouville’s theorem but now for harmonic functions. The authors begin, with a nod to Ed Nelson’s cute proof (see figure below), by showing that a positive harmonic function on the plane is bounded.

Ed Nelson’s six-sentence-long article that appeared in the short-lived Mathematical Pearls section of the Proceedings of the AMS: E. Nelson, A Proof of Liouville’s Theorem.

Nelson’s idea is further developed to prove that sublinear boundedness is enough to guarantee the conclusion. Generalizing Liouville’s theorem in the form described earlier to harmonic functions on \(\mathbb{R}^n\) is moved to the exercises. Notice that this is a sophisticated exercise; the authors follow a route that requires knowledge of the Fourier transform of a tempered distribution. However, it is broken down into small steps with appropriate references. This is a good example of how the authors take the material developed much further in the problem sets. Applications of these two chapters include a proof of Jensen’s formula, Rado’s theorem and two sections on well-known theorems by the pairs of brothers: Frigyes and Marcel Riesz, and Rolf and Frithiof Nevanlinna.

Chapter 14 provides a few more sophisticated applications including Riesz-Thorin interpolation, the Phragmén-Lindelöf principle, and Schwarz’s integral formula. This ends our description of the first fourteen chapters that form the core of the text. Thus far, it appears that not much has changed from the previous edition.

The last six chapters (15 through 20) that make up Fournais-Møller’s addition are devoted to presenting “the complex analysis backbone of Loewner theory” with a view to the “function theoretic backbone of the stochastic Schramm-Loewner equation, which played a key role in the works for which W. Werner (2006) and S. Smirnov (2010) were awarded the Fields medal”. With this preparation the interested student is invited to proceed with their study of Schramm-Loewner theory via Lawler’s Conformally Invariant Processes in the Plane.

The authors start out with a study of the topology of the complex plane that includes a self-contained proof of Jordan’s curve theorem and ends with Schönflies’s theorem on the simplicity of boundary points in Jordan regions. There is a brief chapter on the Dirichlet problem, which is followed by a useful chapter on extending Riemann maps to the boundary. The latter includes a neat exposition of Caratheodory’s theory of prime ends. Next up is Gronwall’s area theorem and some of its geometric consequences (Koebe’s quarter, distortion and growth theorems), with a view to proving Caratheodory’s convergence theorem. The penultimate chapter on logarithmic and half-plane capacities is worked out in some detail, and finally all this spadework is neatly applied to deriving the Loewner equation for the disk, the slit-plane and the half-plane. This is an attractive chapter that ties together many strands of the theory developed thus far.

Classical sources for this material include Duren’s Univalent Functions (Springer, 1983) and Goluzin’s Geometric Theory of Functions of a Complex Variable (AMS, 1969). Another excellent treatment of the extension theory of Riemann maps (and much beyond) is Pommerenke’s Boundary Behaviour of Conformal Maps (Springer, 1992). Finally, an elementary account proving Jordan’s curve theorem is D. E. Sanderson’s 1981 Allendoerfer Prize winning “Advanced Plane Topology from an Elementary Standpoint.”

Tushar Das is an Assistant Professor of Mathematics at the University of Wisconsin-La Crosse.

  • Power Series
  • Holomorphic and Analytic Functions
  • Exponential Function, Logarithm and Winding Number
  • Basic Theory of Holomorphic Functions
  • Global Theory
  • Isolated Singularities
  • The Picard Theorems
  • Geometric Aspects and the Riemann Mapping Theorem
  • Meromorphic Functions and Runge's Theorems
  • Representations of Meromorphic Functions
  • The Prime Number Theorem
  • Harmonic Functions
  • Subharmonic Functions
  • Various Applications
  • Jordan Regions
  • The Dirichlet Problem and Green's Functions
  • Extending Riemann Maps to the Boundary
  • Carathéodory Convergence
  • Capacities
  • Loewner Theory