I'm having trouble understanding the argument principle as seen in Gamelin complex analysis.
I want to go over the example in the text, so I will link the text of the example and go through it.
We want to find the roots of polynomial $z^6+9z^4+z^3+2z+4$ in the first quadrant given by $\{0 < \text{arg} z < \pi/2 \}$. So we consider the quarter disk $D_{R}$ in the first quadrant, consiting of poitns in the first quadrant satisfying $|z| < R$.
The boundary of $D_R$ is given by going through the real axis, then the quarter circle, and then going down the imaginary axis.
We can see that going through the real axis, the change in argument is $0$.
In the quarter circle, the text says that the term $z^6$ dominates, so we just consider $\text{arg} p(z) \approx 6 \text{arg} z$, so the change in argument is $3 \pi$.
My question is, how do we see that $z^6$ "dominates"?
And in the part where we go down the imaginary axis, $z^6$ does not dominate. Why is this so?
Thanks for your help!
Best Answer
"$z^6$ dominates" means that when $|z|$ is "large" you have $$ p(z)\approx z^6 $$ (All other terms $9z^4+z^3+2z+4$ are relatively small compared with $z^6$.) In particular, when $R$ is large, on $\Gamma_R$, you have (1).
To be precise, you have $$ \lim_{|z|\to\infty}\frac{|p(z)|}{|z^6|}=1 $$