Complex Analysis - Maths KU

6.6 Some Consequences of the Residue Theorem 369
Solution Observe that if we identify p(z) =z2 +4, then the three assumptions
(i)–(iii) preceding (37) hold true.The zeros of p(z) are ±2i and correspond
to simple poles of f(z) =π cot πz � (z2 + 4).According to the formula in (41),
∞�
k=−∞
1
k2 �
= −
+4
Res
�
π cot πz
z2 �
, −2i
+4
Now again by (4) of Section 6.5 we have
�
π cot πz
Res
z2 �
, −2i =
+4 π cot 2πi
4i
+ Res
�
π cot πz
z2 ��
, 2i . (44)
+4
�
π cot πz
and Res
z2 �
, 2i =
+4 π cot 2πi
.
4i
The sum of the residues is (π/2i) cot 2πi.This sum is a real quantity because
from (27) of Section 4.3:
Hence (44) becomes
π π cosh(−2π)
cot 2πi =
2i 2i ( −i sinh(−2π))
∞�
k=−∞
= −π coth 2π.
2
1
k2 π
= coth 2π. (45)
+4 2
This is not quite the desired result.To that end we must manipulate the
. Observe
summation � ∞
k=−∞ in order to put it in the form � ∞
k=0
∞�
k=−∞
1
k 2 +4 =
k=−∞
=
�−1
k=−∞
∞�
k=1
k=1
1
k 2 +4 +
k =0
term
����
1
4 +
∞�
k=1
1
(−k) 2 1
+
+4 4 +
∞� 1
=2
k2 1
+
+4 4 =2
∞�
k=1
∞�
k=0
1
k 2 +4
1
k 2 +4
1
k 2 +4
1
− . (46)
4
Finally, we obtain the sum of the original series by combining (45) with (46),
∞� 1
k2 +4 =2
∞� 1
k2 1 π
− = coth 2π,
+4 4 2
and solving for � ∞
k=0 :
∞�
k=0
k=0
1
k2 1 π
= + coth 2π. (47)
+4 8 4
With the help of calculator, we find that the right side of (47) is approximately
0.9104.