CHAPTER 1
Early Triumphs
Nothing illustrates the extraordinary power of complex function theory better
than the ease and elegance with which it yields results which challenged and often
baffled the very greatest mathematicians of an earlier age. In this brief chapter,
we consider two such examples: the solution of the “Basel Problem" of evaluating
∑∞
1
1/n2 and the proof of the Fundamental Theorem of Algebra. To be sure, these
achievements predate the development of the theory of analytic functions; but, even
today, complex variables offers the simplest and most transparent approach to these
beautiful results.
1.1. The Basel Problem
Surely one of the most spectacular applications of complex variables is the use
of Cauchy’s Theorem and the Residue Theorem to find closed form expressions for
definite integrals and infinite sums. As an illustration, we evaluate the sums
ζ(2k) =

n=1
1
n2k
, k = 1, 2, . . . .
The function
H(z) =
2πi
e2πiz 1
is meromorphic on C with simple poles at the integers, each having residue 1, and no
other singularities in the finite plane. It follows that if f is a function analytic near
the point z = n (n Z), then Res(H(z)f(z), n) = f(n). We choose f(z) =
1/z2k
for k fixed and consider the integral
(1.1) IN =
1
2πi
ΓN
H(z)
1
z2k
dz,
where N is a positive integer and ΓN is the positively oriented boundary of the
square with vertices at the points (N + 1/2)(±1 ± i). By the Residue Theorem,
(1.2) IN =
N
n=−N
Res H(z)
1
z2k
, n = Res H(z)
1
z2k
, 0 + 2
N
n=1
1
n2k
.
A routine estimate shows that H is uniformly bounded on ΓN with bound inde-
pendent of N . Thus
H(z)
1
z2k
= O
1
N 2k
on ΓN ;
and since ΓN has length 8N + 4, it follows from (1.1) that
IN = O
1
N 2k−1
.
1
http://dx.doi.org/10.1090/ulect/058/01
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