# problem applying Rouche's theorem for counting zeros

• Apr 17th 2011, 09:02 AM
hatsoff
problem applying Rouche's theorem for counting zeros
Hey guys. I have the following theorem:

Quote:

Originally Posted by Rouche
Let $f$ and $g$ be holomorphic inside and on a contour $\gamma$ and suppose that $|f(z)|>|g(z)|$ on the image $\gamma^*$ of $\gamma$. Then $f$ and $f+g$ have the same number of zeros inside $\gamma$.

I'm supposed to count the zeros of, for example, z^5+15z+1 on D(0;2) (the open disk about 0 of radius 2). But this gives me contradictory results.

Notice that |z^5+1|\geq ||z^5|-1|=31>30=|15z| on the contour $|z|=2$. It follows by Rouche that $z^5+1$ and z^5+15z+1 have the same number of zeros on $D(0;1)$.

But now observe that |z^5|=32>31=15|z|+1\geq|15z+1| on the same contour $|z|=2$ So again by Rouche, $z^5$ and z^5+15z+1 have the same number of zeros on $D(0;1)$.

Putting these together, we see $z^5+1$ and $z^5$ have the same number of zeros on D(0;2). But we can see that $z^5+1$ has five zeros on D(0;2), while $z^5$ only has one zero.

Does anyone have a couple minutes to show me where I've gone wrong?

Thanks!

NOTE: The forum's latex compiler is apparently acting up, so forgive the code.
• Apr 17th 2011, 09:19 AM
Opalg
$z^5$ has a zero of order 5 at the origin, and that counts as five zeros for the purposes of Rouché's theorem. You must always count zeros according to their multiplicity.