Number of zeros in each domain (Complex Analysis)

**shadow_2145** · October 29th 2008, 07:24 PM

Find the number of zeros in each of the domains.

(1) $z^{10} -10z^7 +2iz^4 -z^3 +i$

(a) ${|z|<\frac{1}{2}}$

(b) ${|z|<1}$

(c) ${|z|<2}$

(d) ${|z|<3}$

I have no idea how to approach these. If someone could help me out or point me in the right direction for one of these, I would appreciate it. Thanks!

**Opalg** · October 30th 2008, 03:07 AM

Use Rouché's theorem. In each case, find the dominant term (or terms) for that value of |z|, and apply the theorem.

To take part (b) as an example, when |z|=1 it's clear that $|z^{10} +2iz^4 -z^3 +i|<|-10z^7 |$ . By Rouché's theorem, $z^{10} -10z^7 +2iz^4 -z^3 +i$ has the same number of zeros inside the circle |z|=1 as $-10z^7$ , namely 7.

**whipflip15** · October 30th 2008, 03:15 AM

Very nice.

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