# Counting Zeroes of Complex Functions

• Nov 13th 2010, 12:38 PM
ComplexXavier
Counting Zeroes of Complex Functions
Hello,

How many zeros (counted with multiplicity) does z^4 − 5*z + 1 have in the annulus
{z | 1 < |z| < 2}?

Thanks!
• Nov 13th 2010, 01:09 PM
Drexel28
Quote:

Originally Posted by ComplexXavier
Hello,

How many zeros (counted with multiplicity) does z^4 − 5*z + 1 have in the annulus
{z | 1 < |z| < 2}?

Thanks!

Do you know Rouche's theorem?
• Nov 13th 2010, 09:23 PM
ComplexXavier
Yes, I am familiar with Rouche's theorem, too familiar.

I have tried other questions and fail to find the correct answer and I was just hoping I could see a question done correctly to further my understanding. I appreciate it!
• Nov 14th 2010, 01:12 AM
Opalg
Quote:

Originally Posted by ComplexXavier
Hello,

How many zeros (counted with multiplicity) does z^4 − 5*z + 1 have in the annulus
{z | 1 < |z| < 2}?

Thanks!

The key to applying Rouché's theorem in problems like this is to figure out which term is dominant on each component of the boundary.

On the inner boundary of the annulus, where \$\displaystyle |z|=1\$, the term \$\displaystyle 5z\$ is the dominant one in the function \$\displaystyle z^4 - 5z + 1\$, because \$\displaystyle |5z|=5\$ there (obviously), and that outweighs the size of the other two terms since \$\displaystyle |z^4+1|\leqslant 2\$.

But on the outer boundary of the annulus, where \$\displaystyle |z|=2\$, the \$\displaystyle z^4\$ term is dominant, because \$\displaystyle |z^4|=16\$ there, and that is greater than \$\displaystyle |-5z+1|\$, which is at most 11.

That should enable you to use Rouché's theorem to find how many zeros of \$\displaystyle z^4 - 5z + 1\$ there are inside each of the circles \$\displaystyle |z|=1\$ and \$\displaystyle |z|=2\$, and consequently how many there are in the annulus.