Reviews for Functions of One Complex Variable I

The average rating for Functions of One Complex Variable I based on 2 reviews is 4.5 stars.

Review # 1 was written on 2019-11-11 00:00:00 Rhonda Gilbert For a long time, mathematicians deemed it impossible or imaginary to extract the square root of minus one. Nevertheless, if one did so, one could adopt a fairly consistent manner of computing formally with these imaginary quantities, which are thrust upon one when seeking roots of polynomial equations. Even for polynomials of third degree with real roots, the cubic formula discovered in the sixteenth century requires one to go through imaginary numbers as an intermediary. With the development of algebra into its maturity in the nineteenth century, people became more comfortable about expanding the number system. William Hamilton defined the complex number field in terms of component-wise operations (on the real and imaginary parts); but there are even better algebraic ways to get to complex numbers from the reals, or ultimately from the rationals or just the natural numbers. As Cauchy first realized, contour integrals in the complex plane impart to complex analysis a flavor entirely different from real analysis. During the later course of the nineteenth century, classical function theory blossomed into what is perhaps the most beautiful and profound branch of mathematics ever (this reviewer would be tempted to contradict Gauss, who famously declared arithmetic'i.e., number theory'to be the queen of mathematics! As would anyone with a passion for the continuum and for the infinite, he supposes). Any bachelor in mathematics or physics will take at least a semester of complex analysis, if not more. Unless perhaps one is an engineer, though, an undergraduate preparation is insufficient if one would be a research mathematician. The contour integral should be defined for paths of bounded variation and not, heaven forfend, just for piecewise smooth arcs; the Riemann mapping theorem and Picard's theorem should be proved and not just quoted etc. That is why complex analysis at the graduate level makes up a standard part of the doctoral qualifying examination. Conway's Functions of one complex variable is a staple textbook for a first semester in graduate complex analysis. Another offering at this level would be the redoubtable Ahlfors, which seems to be more computational in orientation, less modern in outlook and possibly slightly less advanced. With Conway, one can be sure of getting an up-to-date point of view in the clean modern notation one comes to expect, if one is a student of higher mathematics these days (post-Bourbaki, that is). Certainly, it is pitched at a more sophisticated level than his undergraduate textbook by Konrad Knopp. The first two chapters start off with the complex number system and basic notions in point-set topology. The exposition is compact, efficient and no-nonsense and designed to cover everything needed later, up to the Weierstrass M-test for uniform convergence. Conway gets underway in earnest in chapter three with the elementary properties of analytic functions and their power series expansions. The closing section of the chapter on Möbius transformations is really quite nice and complete. Chapter four introduces the contour integral, viewed as a Riemann-Stieltjes integral for rectifiable paths, the connection between the contour integral and power series representation, Cauchy's theorem in three versions and Goursat's theorem (the only effective way to show that a function is indeed analytic in a domain). Conway's derivations are clean and modern in format, a pleasure to read. This agreeable style continues to hold in the subsequent chapters on singularities, the maximum modulus theorem, spaces of analytic respectively meromorphic functions, Riemann mapping and Weierstrass factorization theorems, Runge's and Mittag-Leffler's theorems, analytic continuation and Riemann surfaces and finally harmonic respectively entire functions. The last chapter concerns Picard's remarkable little and great theorems on the range of an analytic function. Purists might complain about how Conway mixes general results on metrizable function spaces (such as the Arzela-Ascoli theorem) with those specific to holomorphic functions in the complex plane (Hurwitz's and Montel's theorems, for instance) during the course of his discussion of compactness and convergence in spaces of analytic functions. The sin is not as egregious as one might suppose, though, as it does allow for a streamlined notation. Besides, why complicate matters by jumping off from a starting point of greater generality than will ever be needed in this text? The Riemann mapping theorem to the effect that every simply connected region in the complex plane is conformally equivalent to the unit disk receives a clever and concise proof, as an application of the magic of normal families. This serves as an eye-opener to the power of the deep methods of complex function theory. As an undergraduate, this reviewer was amazed and wondered how it could ever be proved. The Weierstrass factorization theorem follows easily and immediately after a development of the theory of infinite products. The latter also leads to a detailed treatment of the gamma and Riemann zeta functions, as in Euler's representation as an infinite product over the prime numbers. Some may fault Conway's approach to Riemann surfaces via the sheaf of germs of analytic functions on an open set. This reviewer took it in as a nice demonstration of technique. The monodromy theorem, but not existence of the analytic continuation, is shown. Granted, the exposition remains all-too abstract and very little is provided in the way of worked examples. The construction of the Riemann surface for an algebraic equation, for instance, a standard topic at this level, is entirely omitted. Conway's treatment of harmonic functions, on the other hand, is altogether satisfactory and involved, going all the way to subharmonic and superharmonic functions and the solution of the Dirichlet problem and Green's functions. For the most part, the over 440 homework exercises scattered throughout the text test the reader's comprehension of the material. Many turn out not to be so difficult once one unpacks the definitions. Frequently, the exercises serve as a chance to force the reader to supply the derivation of a result, or a step in a proof, that was skipped over in the text. There is, however, a fair sampling of more challenging problems, the solution of which yields a sense of satisfaction. Rarely, however, do they call for heavy computation. Presumably, if the reader wants to try his hand at very computational exercises in complex variables, he could refer to Ahlfors' text. Here, perhaps, we can see the outcome of a difference in perspective; Ahlfors was a great complex function theorist and analyst, while Conway is an operator theorist in his professional life and, thus, feels really at home in the setting of functional analysis with its more abstract point of view. For the beginning graduate student looking to hone his skills in complex analysis in preparation for the qualifying examination, Conway will not disappoint. It's all there; the level reaches uniformly up to a consistent standard, about right for an advanced student who proposes to engage in doctoral research in mathematics. Conway's proofs are clear and complete, with enough detail to follow without being over-fastidious, a good model and exemplary enough of the techniques that the omitted proofs can usually be found out without too much trouble by looking at what Conway does in those included. Many more specialized topics in the theory of a single complex variable are taken up in Conway's companion volume, part II. Part I alone, however, does contain its share of fireworks: the Riemann mapping theorem and Picard's little and great theorems are showpieces of the surprising power of complex analysis (as if the basics of the subject were not enough to impress the onlooker!). The student will have to continue on to a second semester to get to the profoundly beautiful and elegant results that emerge in the theory of prime numbers, elliptic functions, abelian integrals and Riemann surfaces proper.

Review # 2 was written on 2019-10-31 00:00:00 Seth Whitney One of the best introductory books to the theory of complex analysis. The approach is purely analytic with an elite methodology in the discussion of topics. Very beneficial book for graduate students who have basic knowledge to the theory and want to take it further.