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Chapter 10
Discrete Applications of the Residue
Theorem
All means (even continuous) sanctify the discrete end.
Doron Zeilberger
On the surface, this chapter is just a collection of exercises. They are more involved than any of
the ones we’ve given so far at the end of each chapter, which is one reason why we lead the reader
through each of the following ones step by step. On the other hand, these sections should really
be thought of as a continuation of the lecture notes, just in a diFerent format. All of the following
‘problems’ are of a
discrete
mathematical nature, and we invite the reader to solve them using
continuous
methods—namely, complex integration. It might be that there is no other result which
so intimately combines discrete and continuous mathematics as does the Residue Theorem 9.9.
10.1
InFnite Sums
In this exercise, we evaluate—as an example—the sums
°
k
≥
1
1
k
2
and
°
k
≥
1
(
−
1)
k
k
2
. We hope the
idea how to compute such sums in general will become clear.
1. Consider the function
f
(
z
)=
π
cot(
πz
)
z
2
. Compute the residues at all the singularities of
f
.
2. Let
N
be a positive integer and
γ
N
be the rectangular curve from
N
+1
/
2
−
iN
to
N
+1
/
2+
iN
to
−
N
−
1
/
2+
iN
to
−
N
−
1
/
2
−
iN
back to
N
+1
/
2
−
iN
.
(a) Show that for all
z
∈
γ
N
,

cot(
πz
)

<
2. (Use Exercise 31 in Chapter 3.)
(b) Show that lim
N
→∞
±
γ
N
f
= 0.
3. Use the Residue Theorem 9.9 to arrive at an identity for
°
k
∈
Z
\{
0
}
1
k
2
.
4. Evaluate
°
k
≥
1
1
k
2
.
5. Repeat the exercise with the function
f
(
z
)=
π
z
2
sin(
πz
)
to arrive at an evaluation of
²
k
≥
1
(
−
1)
k
k
2
.
104
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View Full Document CHAPTER 10. DISCRETE APPLICATIONS OF THE RESIDUE THEOREM
105
(
Hint
: To bound this function, you may use the fact that 1
/
sin
2
z
= 1 + cot
2
z
.)
6. Evaluate
°
k
≥
1
1
k
4
and
°
k
≥
1
(
−
1)
k
k
4
.
10.2
Binomial Coeﬃcients
The binomial coeﬃcient
±
n
k
²
is a natural candidate for being explored analytically, as the binomial
theorem
1
tells us that
±
n
k
²
is the coeﬃcient of
z
k
in (1 +
z
)
n
. As an example, we outline a proof of
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This note was uploaded on 02/27/2012 for the course MATH 417 taught by Professor Staff during the Fall '11 term at SUNY Albany.
 Fall '11
 Staff

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