# Complex variable ( Rouche's Theorem)

I need some help showing a step-by-step calculation to this question: (better formula representation in attachment):

Astronomer Cal Sagan Discovered from smoke filled telescopic lenses, a planet with billions upon billions of moons call that number N moons, including multiplicity, which he termed "fuzzy" moons. He came up with a 2-dimensional experimentally based model of this with |z|â‰¤1 representing the planet and the location of moons given by the roots of Z^n + a_(N+1) Z^(N-1)..........+ a_1Z + a_0 = 0 in the Complex-plane with |a_0|>1 + |a_(N-1)| + ......... + |a_1| and 2^N> |a_(N-1)|2^(N-1) + |a_(N-2)|2^(N-2) + ......+|a|2 + |a_0|.

He couldn't quite figure out why all of his moons lied in this region. Show using Rouche's Theorem that all the N moons lie in 1 <|Z|>2

https://brainmass.com/math/complex-analysis/complex-variable-rouches-theorem-579636

#### Solution Summary

This solution uses Rouche's Theorem to find the location of roots of a complex polynomial with step-by-step calculations.