FE-P5

[Victor Ivrii]

Consider $P(z)= z^3 +2z -3-i$ and, using the argument theorem and Rouché's theorem calculate the number of its roots in each of the following domains:

(a)  $\{z\colon |z-1|<1\}$;

(b)  $\{z\colon |z-1|>1, |z|<2\}$,

(c) $\{z\colon |z|>2\}$.

[hz12]

P(z)=z^3 -3 + 2z -i = (z^3 – 3)+(2z- i)

a, |z-1|<1
     0 < z < 2
Using Rouche Theorem, let z = 2, z^3-3= 5 > 2z - i, z^3 -3 is the  dominant,
z^3-3 =0 and  z has 3 roots.
So it means when z<2, P(z) has 3 roots.
                                   let z = 0, z^3-3 = -3 < 2z - i, 2z - i is the  dominant,
2z - i =0 and  z has 1 root.
So it means when z<0, f(z) has 1 root.

So when 0 < z < 2, f(z) has 3-1 =2 roots.

b, |z-1|>1, |z|<2
     z < 0, z > 2, -2 < z < 2 so -2 < z < 0
when -2 < z < 0, it has 1 root.

c,  z > 2
 when z < 2, it has 3 roots already, so when z > 2, it has 0 root.

[yujiaha1]

Here's the answer

[Ziqi Zhang]

(a) P(z)=z³+2z−3−i=(z-1)³+3(z-1)²+5(z-1)-i
     at |z−1|=1, |5(z−1)|≥|(z-1)³+3(z-1)²-i|
     so at |z−1|<1, |5(z−1)| and P(z) have the same number of roots which is 1 (z=1).

(b) At |z|=2, |z³|≥|2z-3-i|
     so at |z|<2, |z³| and P(z) have the same number of roots which is 3 (z=0 has the multiplicity of 3).
     but we know |z−1|<1 is in |z|<2, z≠1, so at |z−1|>1,|z|<2, P(z) has the number of roots of 3-1=2.

(c) Because P(z) has the highest power of 3, so it has in all 3 roots, but we already find 3 roots at |z|<2,
     so at |z|>2, the number of roots is 0.

[Victor Ivrii]

(a) As $|z|=2$ consider $Q(z)=z^3$; then  $|Q(z)|=8$,
$$
|P(z)-Q(z)|=|2iz -3-i|\le 4+|3+i|=4+\sqrt{10}<8;$$
therefore $P$ has as many roots in $\{z\colon |z|<2\}$ as $Q(z)$ has, which is $3$ (we count orders).

(c) Then there are no roots in $\{z\colon |z|>2\}$.


(b) Consider $z=w+1$ with $|w|=1$; then $P(w+1)=w^3+3w^2 +5w -i$ and we set $Q(w)=5w$; then $|Q(w)|=5$,
$$
|P(w)-Q(w)|=|w^3+3w^2 -i| < 5.
$$
Indeed, $|P(w)-Q(w)|\le |w+3|+1< 5$ except $w=1$, and for $w=1$ we have $|w^3+3w^2 -i|=|4-i|=\sqrt{17}<5$. Therefore $P(w+1)$ has as many roots in $\{w\colon |w|<1\}$ as $Q(w)$ has, which is $1$ (we count orders).

Finally, in the domain  $\{z\colon |z-1|>1, |z|<2\}$, there are $3-1=2$ roots.