rouche-theorem in problems

1. Estimate number of zeros for polynomial functions in region

  1. Function can be split into a sum of two functions $f(z) + g(z)$.
  2. Estimate the norms $|g(z)|$ and $|f(z)|$ on $\delta D$.
  3. The greater function limits the number of zeros according to the rouche-theorem.

Note1: If region is not simply connected it can be decomposed: problems-using-rouche-theorem-decomposition.png The number of zeros in the left region equals to the sum of zeros counts in regions on the right. So we can estimate number of zeros in simply-connected regions and then calculate the number for not-simply-connected region.

Note2: Also we can use the fundamental-theorem-of-algebra on the $\mathbb{C}$ plane (e.g. for regions ${|z| > 2}$).

Note3: Radial representation of the complex number ($z = |z|\cdot e^{i \varphi}$) is your friend. This can give a better estimate for $|g(z)|$ and $|f(z)|$.