Complex analysis. Fundamentals of the classical theory of functions (Q1031100)

This clear, concise introduction to the classical theory of one complex variable is based on the premise that ``anything worth doing is worth doing with interesting examples.'' All modern introductions to complex analysis follow, more or less explicitly, the pattern laid down in the monograph of \textit{E.~T.~Whittaker and G.~N.~Watson} [A Course of Modern Analysis. Cambridge: University press. (1915; JFM 45.0433.02)]. In ``part I'', we find the foundational material, the basic definitions and theorems. In ``part II'' we find the examples and applications. Slowly we begin to understand why we read part I. Historically this is an anachronism. Pedagogically it is a disaster. Part II in fact predates part I, so clearly it can be taught first. Why should the student have to wade through hundreds of pages before finding out what the subject is good for? In teaching complex analysis this way, we risk more than just boredom. Beginning with a series of unmotivated definitions gives a misleading impression of complex analysis in particular and of mathematics in general. The classical theory of analytic functions did not arise from the idle speculation of bored mathematicians on the possible consequences of an arbitrary set of definitions; it was the natural, even inevitable, consequence of the practical need to answer questions about specific examples. In standard texts, after hundreds of pages of theorems about generic analytic functions with only the rational and trigonometric functions as examples, students inevitably begin to believe that the purpose of complex analysis is to produce more such theorems. We require introductory complex analysis courses of our undergraduates and graduates because it is useful both within mathematics and beyond. Why then do our textbooks create the opposite impression? Ideally part II would repair some of the damage done during part I. Unfortunately the demands of the academic calendar and the ambient mathematical culture have slowly eviscerated part II. Students now are lucky to see even the most elementary properties of the Gamma function and the Weierstrass \(\wp\) function. The fact that these functions are important beyond the confines of complex analysis is scarcely mentioned. What is needed is a change in emphasis and timing. Part II needs to expand again, though perhaps not to the size it had in Whittaker and Watson's days. Part I needs to contract correspondingly. More importantly though, as much as possible of part II must be presented before part I, so that what remains of part I is seen to be genuinely useful. The paragraphs above describe how the author thinks the theory of analytic functions should be presented. What follows is his attempt to implement these ideas. The intended audience for this book is anyone who has taken a calculus course, who knows or is willing to believe the elementary theorems of real analysis given in the appendix, and who wants to learn the classical theory of analytic functions. The pace is rather fast. In compensation the book is quite short. Classical complex analysis is based on a few, very powerful, ideas. They can be presented and illustrated quite quickly if one avoids the many possible entertaining detours along the way. This textbook is divided into three chapters of roughly equal length. It begins with a chapter on the classical special functions. There are several reasons for introducing these from the beginning. One is that these functions arise frequently in many branches of mathematics, pure and applied, and a mathematician should be able, at a minimum, to recognize them and have some idea what is known about them. A second reason is that special functions are a convenient example, often the motivating example, for the general theory. In addition to the Gamma and Beta functions and what are often called the functions of mathematical physics, the author includes sections on the zeta function and its implications for the distribution of primes. All of this material is presented from a `real variable' point of view. This may seem odd in a book on complex analysis, but it is not without reason. Real variable techniques, particularly those depending on convexity and Hölder's inequality, are often useful and worth knowing. They have their limits though, and we soon reach the point where further progress requires the techniques of complex analysis. Complex analysis makes its entrance in the second chapter. This necessarily starts out rather drearily, as the various equivalent definitions of analytic and meromorphic functions are given and their elementary properties proved. In these sections the author willingly sacrifices elegance for speed. Once they are out of the way the serious work can begin. One of the most powerful and, for the novice, most mysterious techniques of complex analysis is Cauchy's calculus of residues. This is in some sense the point of analytic and meromorphic functions. The chapter begins with the easiest examples, rational functions integrated along the unit circle. It concludes with some rather intricate calculations. We hope that these latter give some indication of the enormous power of the calculus of residues. The third part of the book covers elliptic and modular functions. The author treats these in much more detail, and from a different point of view, than is usual in introductory books. The deepest applications of this material are to the theory of numbers. As an example the author gives Eisenstein's proof of biquadratic reciprocity. A proper treatment of complex multiplication would require some number theory, but the author sees at least to provides the necessary analytic background. The book is easy to read and with great interest. It can be recommended to both students as a textbook and to mathematicians and physicists as a useful reference. [See also the review of the original edition (1998; Zbl 0899.30001).]