Construct the Mikhailov hodograph for the equation $$f(z)=z^3+z^2+z+2$$
Here's my solution:
We have $$f(i\omega)=(-\omega^2+2)+i(-\omega^3+\omega)$$.
We consider $Ref(i \omega)=0$ and $Re \omega >0 \implies \omega_2=\sqrt{2}$
We consider $Imf(i \omega)=0$ and $Re \omega >0 \implies \omega_0=0, \omega_1=1$.
- Whence, $f(i\omega_0)=2$, $f(i\omega_1)=1$, $f(i\omega_2)=-i\omega_2$
1/ Now, I have trouble when I construct the Mikhailov hodograph (starts on the positive real semi-axis, does not hit the origin and successively generates an anti-clockwise motion through $n$ quadrants.
(An equivalent condition is: The radius vector $f(i \omega)$, as $\omega$ increases from $0 $ to $+\infty$ , never vanishes and monotonically rotates in a positive direction through an angle $\dfrac{n\pi}{2}$), anyone can describes a curve? Thanks!
2/ I'm trying to find $\phi$ from $\phi=\dfrac{\pi}{2}(n-2m);n=3$. I have a question $\phi=?$. I mean How we can find $\phi$? Why? Thanks!