Page 75 - Calculus with Complex Numbers

We let yx be the square centre 0 with half side (N + 1/2):v (Figure 7. 1), and
consider the integral

cot z
2 Jz.
y,v Z
The integrand has singularities at z = n:r where the residues are

cot z 1
Res 2 =
s2n.2

for n # 0. At z = 0 the Laurent expansion is

showing that there is a triple pole at z = 0 with residue - 1/3.
Therefore by the residue theorem (see Section 4.7) we have

/-

lf we can show the integral --> 0 as N --> co then we get

and hence

oo 1 :a.2
X
ky - o .
1