Page 100 - Calculus with Complex Numbers

which on letting N --> co gives

2 * 1 2
O = - V - -,
zr z'- rltrl + 1) zr
1
and hence

8 Fundam ental theorem of algebra

Therefore by Rouché's theorem f (z) = z3, f (z) + g (z) = .:3 + 9z2 + 9.z + 9

:
have the same number of zeros inside I z I= 10. But f (z) = .3 has 3 zeros

inside Izl = 10, all at z = 0. Hence also .3 + 9z2 + 9.z + 9 has 3 zeros inside
:
I z I = 10, as required.

Thereforeby Rouché's theorem flz) = 9:2, flz) q-g (c) = .z3 +9c2 +9c+9
have the same number of zeros irlside Iz I = 2, namely 2, since f (z) = 9z2

has 2 zeros irlside Izl= 2, both at z = 0.

lf f (z) EEE 9, g (z) = .:3 + 9z2 + 9c, then for all Izl= 1/2 we have

Therefore by Rouché's theorem f (z) EEE 9, flz) + g (z) = .:3 + 9z2 + 9.z + 9
have the same number of zeros inside I zl = 1/2 namely none, since f (z) EEE 9

has no zeros inside IzI= 1/2.
On IzI= 1 we have

IzS + 6zI s IzI2 + 6IzI = 7 < 8.
:
Therefore .3 + 6 : + 8 has no zeros inside Iz I = 1.

.
On I z I= 3 we have