# Application of Rouche's theorem

• Jun 16th 2010, 12:47 PM
brouwer
Application of Rouche's theorem
I want to estimate the number of zeros for the function, by using Rouche's theorem.

\$\displaystyle f(z)=z^2e^z-z\$ in the region \$\displaystyle D(0,2)\$, which is a disc around 0 with radius 2.

I can only find and perform easy examples for polynomials, this one has been bugging me for a while now. No clue in what direction I have to look to pick a function to compare \$\displaystyle f(z)\$ with.
• Jun 16th 2010, 12:58 PM
Ackbeet
Hmm. I would break up your domain into two pieces, depending on whether the imaginary part of \$\displaystyle z\$ is negative or non-negative. One obvious root is the origin. In the right half-plane intersected with your domain, the exponential function makes the \$\displaystyle z^{2}\$ larger, does it not? And smaller in the left-half plane intersected with your domain, correct? Just an idea or two.
• Jun 17th 2010, 11:51 AM
brouwer
Because LaTeX isn't fully working yet, the function mentioned is f(z) = z^2 * e^z - z on the open disc around 0 with radius 2.

@Ackbeet, thanks for the reply but I want to make use of Rouche's theorem.
• Jun 17th 2010, 11:59 AM
Ackbeet
brouwer, I was trying to help you use Rouche's theorem. As stated in Gamelin, p. 229, Rouche's theorem runs like this:

Let D be a bounded domain with piecewise smooth boundary D'. Let f(z) and h(z) be analytic on D union D'. If |h(z)| < |f(z)| for z in D', then f(z) and f(z) + h(z) have the same number of zeros in D, counting multiplicities.

I was trying to help you find function h(z). I was suggesting that you break up your domain into two pieces and apply Rouche's theorem on each piece.