Page 80 - Calculus with Complex Numbers

for Iy I k: n'. And for .x = Nn' we have

7.7 U se of cotzr.z
Corlsider the series

oo 1
X z - 1 .
n
+
1
The integral

cot z
dz ,
ï' . :2 + 1
N
where yx is the square centre 0 with half side (N + 1/2):v will sum the series

* 1
X zzol
1
which is not quite what we want. lnstead we use

cot n'z
dz ,
. :2 + 1
ï'N
where yx is the square centre 0 with half side (N + 1/2) (Figure 7.3).
The singularities of cot n'z occtlr at z = n with residues

coszrz coszrz 1
Res cotzrz = Res = = - .
z=n z=n sinzrc zr coszrz C = / zr

Ffgure 7.3