11. In this exercise, we explore how residues may be used to sum certain series. To begin, let P and Q be polynomials such that deg Q ≥ 2 + deg P , g = P/Q, and f (z) = g(z) cot πz. Note that f is...

11. In this exercise, we explore how residues may be used to sum certain series. To begin, let
P
and
Q
be polynomials such that deg
Q

≥
2 + deg
P
,
g
=
P/Q, and
f
(z) =
g(z) cot
πz. Note that
f
is defined on C except for the isolated singularities Z
∪

Z(Q).

2

(a) Show that there exist
M

>
0 and
R

>
0 such that
|
g(z)|

≤

M
/
|
z
|

such that
|
z
|


>

R.

for all
z

∈
C

(b) For
N

∈
N with
N


> R, let
C
N
denote the rectangular contour with sides lying on the lines Re
z
=
±(N
+ 1/2) and Im
z
=
±
N
. Show that

r

lim

N
→∞

f
(z)
dz
= 0

C
N

by parameterizing the four separate sides of
C
N
and showing their moduli each go to 0.

(c) Verify that for all
n

∈
Z
\

Z(Q), Resz=n

f
(z) =
g(n)/π.

(d) Conclude that

∞

n=−∞

Res
f
(z) =

a
∈
Z(Q)\Z

Res
f
(z).

z=a

(This includes arguing that the series on the left-hand side converges. The right-hand side is understood to be 0 if
Z(Q)
⊆
Z.)

(e) Use parts (c) and (d) and the given
g
to verify the summation identity.

1

i.
g(z) =
, z2

1

n=1

=
π

1

ii.
g(z) =
,

c

>
0,

1 =
π

coth
π
c


1

z2 +
c2

n=1

n2 +
c2 2c

2c2