Page 69 - Calculus with Complex Numbers

Ffgure 8.4

Hence we have

dt + J dz
2
/
t3 + 1 y .ztj .y ) ,
which shows that
oo dt 2zr/ 2:/
Collverges = = ,
o t3 + 1 3* (1 - û?) 3.$ 77
,
,

6.7 H alf residue theorem

To evaluate the integral

co sju x
- co A
we would like to consider the contotlr integral

d iz
- Jc
v C

round the usual D-shaped contour as in Section 6.4 except that the integrand has a
singularity at z = 0. W e use instead the indented contour y = n + yz + p + y4
shown in Figure 6.5.
On n We have z = t (r :jq t :jq A) . Therefore