Use Rouche’s Theorem to show that all roots of $z ^6 + (1 + i)z + 1 = 0$ lines inside the annulus $ \frac{1}{2} \leq |z| \leq \frac{5}{4}$
Answer
Rouche?s Theorem states that if $f(z)$ and $g(z)$ are both analytic inside and on the closed contour $C$, and if $|g(z)| < |f(z)|$ on $C$ then $f(z)$ and $f(z)+g(z)$ have the same number of zeros inside $C$.
i) Let $f(z) = 1, g(z) = z^6 + (1 + i)z$. Then for $|z| = 1/2$
$|g(z)| ? ( 1/2 )^ 6 + 1/2 \sqrt{2} < 1 = |f(z)|$.
Since f(z) has no zeros inside $|z| = 1/2$, and so $z^6 + (1 + i)z+1=f(z) + g(z)$ has no zero inside $|z| = 1/2$ .
ii) Let $f(z) = z^6$ , $g(z) = (1 + i)z + 1$. Then for $|z| = 5/4$ ,
$ |g(z)| ? \sqrt{2}(5/4) + 1 ? 2.77$ < |f(z)| = ( 5/4 )^6 ? 3.81$.
Since $f(z)$ has six zeros inside $|z| = 5/4$, and so $z^6 + (1 + i)z+1=f(z) + g(z)$ has all its six zeros inside $|z| = 5/4$.
The answer is accepted.
- answered
- 151 views
- $10.00
Related Questions
- Integral of trig functions
- Vector field
- Two calculus questions
- Trying to solve this system of simultaneous equations. A solution with work shown would be appreciated.
- Fermat's method of calculus
- Calc 3 Question
- Is $\int_0^{\infty}\frac{x+3}{x^2+\cos x}$ convergent?
- Find the limit as x-->0 for y = (e^x- 1)/[sin(nx)]