This blog post is on Rouche’s Theorem and some applications, namely counting the number of zeroes in an annulus, and the fundamental theorem of algebra.
Rouche’s Theorem: Let , be holomorphic inside and on a simple closed contour , such that on . Then and have the same number of zeroes (counting multiplicities) inside .
Rouche’s Theorem is useful for scenarios like this: Determine the number of zeroes, counting multiplicities, of the polynomial in the annulus .
Let be the unit circle . We have
Since has 2 zeroes in , therefore has 2 zeroes inside , by Rouche’s Theorem.
Let be the circle
on . Therefore has 5 zeroes inside .
Therefore has 5-2=3 zeroes inside the annulus.
We do a computer check using Wolfram Alpha (http://www.wolframalpha.com/input/?i=2z%5E5-6z%5E2-z%2B1%3D0). The moduli of the five roots are (to 3 significant figures): 0.489, 0.335, 1.46, 1.45, 1.45. This confirms that 3 of the zeroes are in the given annulus.
Fundamental Theorem of Algebra Using Rouche’s Theorem
Rouche’s Theorem provides a rather short proof of the Fundamental Theorem of Algebra: Every degree n polynomial with complex coefficients has exactly n roots, counting multiplicities.
Proof: Let . Chose sufficiently large so that on the circle ,
Since has roots inside the circle, also has roots in the circle, by Rouche’s Theorem. Since can be arbitrarily large, this proves the Fundamental Theorem of Algebra.