Preface

At the middle of the twentieth century, the theory of analytic functions of a

complex variable occupied an honored, even privileged, position within the canon

of core mathematics. This “particularly rich and harmonious theory," averred

Hermann Weyl, “is the showpiece of classical nineteenth century analysis."1 Lest

this be mistaken for a gentle hint that the subject was getting old-fashioned, we

should recall Weyl’s characterization just a few years earlier of Nevanlinna’s theory

of value distribution for meromorphic functions as “one of the few great mathemat-

ical events in our century."2 Leading researchers in areas far removed from function

theory seemingly vied with one another in aﬃrming the “permanent value"3 of the

theory. Thus, Clifford Truesdell declared that “conformal maps and analytic func-

tions will stay current in our culture as long as it lasts";4 and Eugene Wigner,

referring to “the many beautiful theorems in the theory ... of power series and of

analytic functions in general," described them as the “most beautiful accomplish-

ments of [the mathematician’s] genius."5 Little wonder, then, that complex function

theory was a mainstay of the graduate curriculum, a necessary and integral part of

the common culture of all mathematicians.

Much has changed in the past half century, not all of it for the better. From

its central position in the curriculum, complex analysis has been pushed to the

margins. It is now entirely possible at some institutions to obtain a Ph.D. in

mathematics without being exposed to the basic facts of function theory, and

(incredible as it may seem) even students specializing in analysis often fulfill degree

requirements by taking only a single semester of complex analysis. This, despite

the fact that complex variables offers the analyst such indispensable tools as power

series, analytic continuation, and the Cauchy integral. Moreover, many important

results in real analysis use complex variables in their proofs. Indeed, as Painlevé

wrote already at the end of the nineteenth century, “Between two truths of the

real domain, the easiest and shortest path quite often passes through the complex

1Hermann

Weyl, A half-century of mathematics, Amer. Math. Monthly 58 (1951), 523-553,

p. 526.

2Hermann

Weyl, Meromorphic Functions and Analytic Curves, Princeton University Press,

1943, p. 8.

3G.

Kreisel, On the kind of data needed for a theory of proofs, Logic Colloquium 76, North

Holland, 1977, pp. 111-128, p. 118.

4C.

Truesdell, Six Lectures on Modern Natural Philosophy, Springer-Verlag, 1966, p. 107.

5Eugene

P. Wigner, The unreasonable effectiveness of mathematics in the natural sciences,

Comm. Pure Appl. Math. 13 (1960), 1-14, p. 3.

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