Problem 7

Exercise 3.7 (Rouché's Theorem on a Geometric Progression)  

1. State Rouché's theorem.
2. Find the number $Z_n$ of zeroes of $p_n$ as $n\to\infty$ within the closed contour $C=\partial B_{1}(1/2)$ where

   $$p_n(z) = z^2 - 2\left(\frac{z}{3} + \cdots + \frac{z^n}{3^n}\right)$$ (3.48)

Theorem 4 (Rouché's Theorem)   Suppose $h = f+g$ where $f$ and $g$ are holomorphic on the interior of some closed contour $C$ and moreover that $\vert f(z)\vert>\vert g(z)\vert$ on the contour $C$. Then $f$ and $h$ have the same number of zeros in the interior of $C$.

Now begin the problem.

Proof. Note that $p_n(z)$ converges uniformly on the set disk $D=\overline{B_1(1/2)}$ to

$$p(z) = z^2 - 2\frac{z/3}{1-z/3} = z^2 - \frac{2z}{3-z}.$$ (3.49)

Solve for the zeroes:

$$0=z^2 - \frac{2z}{3-z} = z^2(3-z) - 2z = z\left[z(3-z)-2\right] = z[-z^2 + 3z - 2] = -z(z-1)(z-2)$$ (3.50)

Only two of the roots $z=0$ and $z=1$ lie inside the contour $C$. None of the roots lie on the contour, which indicates $m=\min\vert z^2-2z/(3-z)\vert>0$ where the minimum is taken over $C$.

Now we can apply the Rouché theorem with $h=p$ as defined above, $f=p_n$ and $g=p-p_n$. On the contour $C$, let us verify the inequality. Let $N$ be such that

$$\vert g(z)\vert = \vert p(z)-p_n(z)\vert = \left\vert\sum_{k=n+1}^\infty \frac{z^k}{3^k}\right\vert < m/2 n\geq N z$$ (3.51)

Then

$$\vert f(z)\vert = \left\vert z^2 - 2\left(\frac{z}{3} + \cdots + \frac{z^n}{3^n}\right)\right\vert$$ (3.52)

$$= \left\vert z^2 - 2\left(\frac{z}{3-z} - \sum_{k=n+1}^\infty \frac{z^k}{3^k}\right)\right\vert$$ (3.53)

$$= \left\vert z^2 - \frac{2z}{3-z} + \sum_{k=n+1}^\infty\frac{2z^k}{3^k}\right\vert$$ (3.54)

$$\geq \vert\vert z^2-2z/(3-z)\vert - \vert\Sigma\vert\vert$$ (3.55)

$$> m/2$$ (3.56)

Since $\vert g(z)\vert<m/2$ and $\vert f(z)\vert>m/2$, this shows that Rouché's theorem applies, indicating $p_n$ and $p$ have the same number of roots, namely 2, inside the contour $C$. $\qedsymbol$