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