BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

Complex analysis texts written in the early 1900’s, mostly by British authors such as E. T. Copson [2] and E. T. Whittaker and G. N. Watson [4], devote considerable space in later chapters to what we call special functions. These include the gamma, the hypergeometric, the elliptic, the modular, and the zeta functions. Paul Halmos [3] wrote an interesting discussion illustrating the neglect of these chapters on special functions when he was a graduate student at Illinois during the 1930’s. Concerning Whittaker and Watson’s book, Halmos writes, It has two parts: “The process of analysis” and “The transcendental functions”. The second part (345 pages) is longer than the first, and I find it frightening. Its twelve chapters have titles such as “The zeta function of Riemann” (I am not yet trembling), “The confluent hypergeometric function” (now I am), and “Ellipsoidal harmonics and Lame’s equation” (I am ready to flee to cohomology and ask for asylum). In the complex variable courses at Illinois in the 1930’s Whittaker and Watson was frequently used, but, to the best of my knowledge, the second part was never entered. The trend observed by Halmos continues today, with special functions receiving scant attention, and modern texts reflect this. In the past two decades, a large number of complex analysis texts have been written. These cover many interesting topics, but not special functions, with the exception of the gamma function or the zeta function. Now, why should special functions be studied? The theory of special functions, developed by some of the greatest names in the history of mathematics and further studied by outstanding contemporary researchers, are essential to solving many important problems in mathematics and mathematical physics. Sometimes the applications of these functions can be surprising or unexpected; a dramatic recent example is de Branges’s solution of the Bieberbach conjecture where the positivity of a specific hypergeometric function turned out to be important. In fact, the properties of special functions can effectively motivate an account of complex analysis while at the same time providing tools which would be useful to a student of mathematics. It is good to see that Stalker’s book reinstates special functions into the teaching of complex analysis by presenting such a well motivated account. His insights have also allowed him to put his presentation into a more logical order. Stalker explains his reasoning in his preface: In “part I” [of Whittaker and Watson] we find the foundational material, the basic definitions and theorems. In “part II” we find 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