Hey guys. I have the following theorem:

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.Originally Posted byRouche

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

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

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