Page 68 - Calculus with Complex Numbers
P. 68
Ffg ure 6.3
by considering the contour integral
/ s
+
dz l
/
round y as in Section 6.2 since the integrand has a singularity at z = - 1 which is
on y . lnstead we use the pizza slice contour y = n + yz + p shown in Figtlre 6.3.
Here û? = el=i/3 is the primitive cube root of unity (see Section 1. 10).
On n We have z = t (0 s t :é A). Therefore,
dt
t3 + 1 '
On p We have z = a)t (R k: t k: 0). Therefore,
dz R û? dt R dt
= - = -u? (u?3 = 1).
n . :3 + 1 () œ3t3 + 1 () t3 + 1
On yz W e have
-
R
l z3 1 Y 3
dz 2>4/31 -* 0
+
as R --> co.
The integrand has a singularity inside y at z = ei=/3 = -û?2 (if R > 1) where
the residue is
Therefore
dz 2>ï
v . :3 + 1 3*
