Page 72 - Calculus with Complex Numbers
P. 72
Example Consider the integral
-
/- -
co (sin-vjz vy.
To evaluate this integral we observe that
oo i 2 1 oo 1 - cos 2.x 1 * 1 - elix
s n .x
dx = - dx = - Re dx,
. x 2 .x2 2 . x2
and work with
1 1 - eliz
- dz
2 . ::2
p'
where y = n + yz + yg + y4 as in Figure 6.5.
The Laurent expansion of the integrand at z = 0 is
1 1 - eliz 1 1 - (1 + liz + . . . ) i
c
i :2 = i c2 = - - V . . . y
which shows that z = 0 is a simple pole with residue -ï . Therefore the half
residue theorem applies and continuing as in the lirst example we obtain
x
oa ( sin.v 2dx - '''
-
.1* . )
c
o
Exam ples
Hvaluate the following integrals.
CO dx
. x2 + 4
(>/6)
(2,,/...,4)
* cos .x dx
tzrc-l cos 1)
. x2 + 2 .x + 2
