Taking Cauchy’s Formulas to the Limit

Question

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch 5.3 Taking Cauchy’s Formulas to the Limit

1. On proving Prop 5.10

Here's what I did. Is this right?

At the end we got

$$0 = \lim_{z \to \infty}||p(z)|-|a_dz^d||$$

$$\iff \forall \varepsilon > 0, \exists R_1 > 0: |z| \ge R_1 \implies ||p(z)|-|a_dz^d|| \le \varepsilon$$

Choose $\varepsilon = \frac 1 2$. Then $\exists R_1 > 0: |z| \ge R_1 \implies ||p(z)|-|a_dz^d|| \le \frac 1 2,$ that is,

$$-\frac 1 2 \le |p(z)|-|a_dz^d| \le \frac 1 2$$

$$\implies -\frac 1 2 + |a_dz^d| \le |p(z)| \le \frac 1 2 + |a_dz^d|$$

Now I'll show $\exists R > 0:$

$$-\frac 1 2 |a_dz^d| \stackrel{(2)}{\le} -\frac 1 2 + |a_dz^d| \le |p(z)| \le \frac 1 2 + |a_dz^d| \stackrel{(3)}{\le} 2|a_dz^d|$$

(2) $\iff R_2:= \frac{1}{|a_d|} \le |z|^d$

(3) $\iff R_3:= \frac{1}{3|a_d|} \le |z|^d$

$$\therefore, R := \max\{R_1,R_2,R_3\}$$

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2. Quick question on application of Fundamental Thm of Algebra 5.11

What is the corollary being referred to? I think we apply Fundamental Thm of Algebra 5.11 again. Is Fundamental Thm of Algebra 5.11 seen as a corollary of Prop 5.10?

Answer 1

For first red box.

Hint: Use $$|1+z_1+\cdots+z_n|\geq1-|z_1|-\cdots-|z_n|\to1$$

For second red box.

when you have a root a from theorem, then apply the theorem again for p/(z−a).