Page 67 - Calculus with Complex Numbers
P. 67
Ffgure 8.2
where y = n + yz as in Section 6.2. The integrand has singularities at the solutions
of the equation ./ + 1 = O which are û?, 0)3 û)5 û)7 (Figure 6.2) where û? = ei=/4
is the primitive 8t.11 root of unity (see Section 1. 10).
Differentiating the denominator we lind the residues of the integrand at z =
û?, 0)3 are 1/4*3 1/4*9 respectively. Therefore,
dz 1 = ln'i 4u?3 + 4.9 = (2.9 (û? + 1) =
,
.
1
=
+
1
j c4 1 1 n'i (5 n'i ( 1 - i ) zr (1 + i )
,
.
,
,
v
since *6 = -ï . nut û? = (1 + illp/'I so we get
as R --> co. Therefore
R dx dz dz dz =
= = - ->
- /1 . :4 + 1 / . / + 1 /2 . / + 1 .vC
R .x4 + 1
as R --> co, which shows that
OD dx >
Converges = .
w 4 + 1 X''j
6 .6 S i ngu I ari ti e s o n the real axi s
W e cannot evaluate the integral
O dx
0 . x3 + 1
