Problem 7

Exercise 2.7 (Complex Fundamental Theorem of Algebra)   State Rouché's theorem and prove the fundamental theorem of algebra.

Theorem 2 (Rouché's Theorem)   If $h = f+g$ and

$$\vert f\vert>\vert g\vert$$ (2.37)

on the contour $C$, then $h$ and $f$ have the same number of roots inside $C$.

Theorem 3 (Complex Fundamental Theorem of Algebra)   Prove that a polynomial

$$P(z) = \sum_{k=0}^n a_kz^k$$ (2.38)

has exactly $n$ roots and the radius of the disk about zero containing all the roots may be estimated.

Proof. Reduce the polynomial to a monic

$$p(z) = \sum_{k=0}^n c_kz^k$$ (2.39)

where $c_n=1$ by dividing by $a_n$. Select

$$R > \sum_{k=0}^{n-1}\vert c_k\vert$$ (2.40)

Then for $\vert z\vert=R$, we have

$$\vert p(z) - z^n\vert = \vert c_{n-1}z^{n-1} + \cdots + c_1z + c_0\vert$$ (2.41)

$$\leq \vert c_{n-1}\vert\vert z\vert^{n-1} + \cdots + \vert c_1\vert\vert z\vert + \vert c_0\vert$$ (2.42)

$$= \vert c_{n-1}\vert R^{n-1} + \cdots + \vert c_1\vert R + \vert c_0\vert$$ (2.43)

$$\leq \vert c_{n-1}\vert R^{n-1} + \cdots + \vert c_1\vert R^{n-1} + \vert c_0\vert R^{n-1}$$ (2.44)

$$< R^n = \vert z\vert^n$$ (2.45)

Then taking $h=p(z)$, $f=z^n$ and $g=p(z)-z^n$ in Rouché's theorem, we see that $p(z)$ and $z^n$ have the same number of zeroes inside the disk of radius $R$ about the origin.

We scaled the polynomial to be monic, so when we unscale it, we can see all the roots lie in the disk of radius $R\vert a_n\vert$ about the origin. $\qedsymbol$