The author introduces the study of quasiconformal mappings as natural generalizations of conformal mappings, as mappings less rigid than conformal mappings, as mappings important in the study of elliptic partial differential equations, as generating interesting extremal problems, as important in moduli theory and Fuchsian and Kleinian groups, and as mappings that are better behaved in the context of several complex variables. This short book gives a general overview of the important properties of quasiconformal mappings, with these goals and properties in mind.

As the author notes, quasiconformal mappings were introduced by the mathematician H. Grotzsch in 1929 as mappings that are as close to conformal as possible, but that can, for example, take a square in the plane and map it to a (non-square) rectangle in the plane, mapping vertices onto vertices in the process. Conformal maps cannot do this, and a measure of how far the quasiconformal map is from being conformal involves the calculation of the dilatation at a point. The dilatation is computed by considering the effect that the linearization of a C1 homeomorphism has on circles. Such circles are mapped to ellipses, and the ratio of the major to the minor axis gives the dilatation. The dilatation is 1 for a conformal mapping. Following Grotzsch, a mapping is then said to be quasiconformal if the dilatation is bounded, and K-quasiconformal if the dilatation is less than or equal to K.

For a family C of curves in the plane, the author introduces the extremal length of C, and shows that it is invariant under conformal mappings and "quasi-invariant" (multiplied by a bounded factor) under quasiconformal mappings. And following precedents from harmonic analysis, the author defines the Dirichlet integral, and shows it to be quasi-invariant under quasiconformal mappings.

A more general definition of a quasiconformal mapping is then given, that relaxes the Grotzsch requirement that the mappings be C1. A (topological) mapping is called K-quasiconformal if the "modules" of quadrilaterals are K-quasi-invariant. Quadrilaterals are defined to be Jordan regions with a pair of disjoint arcs on the boundary. The "module" is then defined to be the ratio of the sides of the rectangle that the quadrilateral is mapped to conformally. A 1-quasiconformal mapping is shown to be conformal under this new definition. This "geometric" definition of quasiconformal is followed by an "analytic" one, namely that a (topological) mapping is K-quasiconformal if it is "absolutely continuous on lines", with another condition that the ratio of its complex derivatives must be bounded by (K-1)/(K+1). A function on a region is "absolutely continuous on lines" if it is absolutely continuous on almost every horizontal and vertical line of any rectangel in the region. The author proves that the geometric and analytic definitions are equivalent.

The author then considers three different extremal problems for a doubly-connected region in a finite plane. The problem of finding the largest value of the module of this region is considered for three different conditions on this region and its bounded and unbounded components. Simple relations are shown to exist between the three different extremal modules. Also, and very interestingly, the author gives a connection between these extremal modules and elliptic curves. Mori's theorem is also proved, which gives a "Holder inequality", i.e. information on the "stretching" ability of K-quasiconformal mappings of the open unit disk to the open unit disk. This then allows a homeomorphic extension to the closed disk. The author then shows that K-quasiconformal mappings of the unit disk unto itself form a sequentially compact family with respect to uniform convergence. He then considers the question as to when quadruples of distinct complex numbers can be mapped to each other under K-quasiconformal mappings.

These results naturally set up a study of the boundary value properties of quasiconformal mappings. The author proves an "M-condition" that is satisfied by a boundary correspondence mapping. This condition involves the parameter K, and the author shows that the boundary values of a K-quasiconformal mapping must satisfy this condition. He then studies the extent to which quasiconformal mappings are isometries, called "quasi-isometries" where distances are multiplied by a bounded factor. The author then gives an interesting discussion as to the geometric properties of curve that the real line gets mapped to when a quasiconformal mapping of the entire complex plane is considered.

Attention is then concentrated on proving the existence of quasiconformal mappings with a given complex dilatation. This entails a solution of the Beltrami equation, and for this purpose the author constructs two different integral operators, one of which acts on Lp functions for p > 2, the other acting on C2 functions with compact support. All of his discussion has the flavor of classical harmonic analysis, with detailed attention given to the proof of the Calderon-Zygmund inequality.

The author ends his book with a consideration of Teichmuller spaces, with attention given to the extent to which the classical theory of conformally equivalent Riemann surfaces can be generalized to the context of quasiconformal mappings. The Teichmuller space is thus defined as the collection of pairs of Riemann surfaces and sense-preserving quasiconformal mappings between them, with pairs being equivalent under homotopy. The familiar Beltrami and quadratic differentials make their appearance, and the image of the mapping of the unit ball of Beltrami differentials to the spece of quadratic differentials is shown to be open. The Teichmuller space is then shown to be an open subset of the space of quadratic differentials, with the Teichmuller metric giving the same metric as the norm in the space of quadratic differentials. This space has dimension 3g - 3 when the starting Riemann surface has genus g, and with respect to the parameters of the basis of this space, one obtains a holomorphic family of Riemann surfaces.

I know this book is a classic, but I don't believe that the approach to the subject is the best possible. First of all, the author never really defines the set of complex number (either as RxR with certain operations or as a splitting field of R[x]). The book is very compact but fails to give the reader first learning complex analysis a good impression of what is going on, having too few and too theoretical examples. I believe that it can be really useful as a reference, but I wouldn't recommend it to someone who is foreign to the subject.

This is a classic complex analysis text, a pleasure to read and covering all the usual topics. The prerequisites are modest; ideally, one will be familiar with the material in Rudin's "Princples of Mathematical Analysis," but a good, mathematically oriented calculus course (Spivak's "Calculus" is beautiful) is quite sufficient.

That said, the price tag is ridiculous. It was bad enough at $90 (judging by previous reviewers, that was back in the ancient days of 2001). The last edition of this book is dated 1979. It's used in graduate courses all around the world. That means that used copies are not hard to come by.

For $143, one can buy a used copy of Ahlfors, and *new* copies of Conway's and Needham's complex analysis books, and still have pocket change left. That's the course I would recommend.

This book has been, since its first edition in 1953, the standard textbook for rigorously learning complex analysis, and not without a reason. The wonderful theory of this branch of mathematics is appropriately emphasized and thoroughly constructed, leading to more general and precise results than most textbooks. While the constant appearance of new texts on the field can only help appreciate the subject from a different perspective, few give you such a deep and serious treatment like this gem.

Postscript: An earlier reviewer claims that Ahlfors never defines the set of complex numbers, while this is indeed done in the fourth through sixth pages in a much more analytical way than generally found elsewhere. It is quite possible to dislike this author's style or approach (or anybody's for that matter), but it would be difficult to charge Ahlfors with being sloppy with his writing.