Page 86 - Calculus with Complex Numbers
P. 86
yz = circle centre 0 radius 2.
We can parametrise yz as z = leit (0 :jq t :jq 2:7r). Therefore, the image contour
f (yz) parametrises as w = f (z) = z2- 1 = 4elit - 1 which is the circle centre - 1,
radius 4 described twice. Hence in this case the image contour f (yz) circulates
the origin twice, rellecting the fact that flz) has 2 zeros inside yz (Figure 8.2).
Case 3 p = circle centre 1 radius 1.
which shows w + 1 = reio where r = 4 cos2 t(l, 0 = t . Hence in this case the
image contour is the cardioid illustrated in Figure 8.3 which circulates the origin
:
once, in agreement with the fact that .2 - 1 has one zero irlside y at z = 1 .
@
Ffg ure 8.2
Ffg ure 8.3
