The Zeros of an Analytic Function

**shadow_2145** · November 16th 2008, 09:41 AM

Determine how many zeros of the given function lie in the given annulus.

(1) $z^3-3z+1$ in $1<|z|<2$

(2) $z^4 -2z -2$ in $\frac{1}{2} <|z|< \frac{3}{2}$

Thanks for any help!

**Opalg** · November 16th 2008, 10:15 AM

Originally Posted by shadow_2145

Determine how many zeros of the given function lie in the given annulus.

(1) $z^3-3z+1$ in $1<|z|<2$

Intermediate value theorem: the function has three real zeros, in the intervals (-2,-1), (0,1), (1,2).

Originally Posted by shadow_2145

(2) $z^4 -2z -2$ in $\frac{1}{2} <|z|< \frac{3}{2}$

Rouché's theorem: $|2(z+1)|<|z^4|$ on |z| = 3/2; $|z^4-2z|<2$ on |z| = 1/2.

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