I need to determine the number of zeros in the right half plane Re z>0 of the polynomial:
$$
f(z)=z^3-z+1
$$
My attempt to solve the problem:
I'm using Rouché's theorem and consider
$$
g(z)=z^3+1 ~~ and ~~
p(z)= -z
$$
and it can be seen that $|g(z)|>|p(z)|$ on the imaginary axis and for $|z|$ large.
My conclusion is that the function $f(z)$ has $3$ zeros in the right half plane. Is this correct?
(A solution from the book showed that it has $2$ zeros in $Re\, z>0$ but I'm not sure if it's a typo)
Best Answer
You correctly derived that $f$ and $g$ have the same number of zeros in the right half-plane.
Only the final conclusion is wrong: $g(z) = z^3+1$ has two zeros in the right half-plane (the third zero is $z=-1$).