MAT334--2020S > Chapter 3

3.1 question 8


Yan Zhou:
$$2z^4 - 2iz^3 + z^2 + 2iz -1$$
find roots for upper half line.
here is the link for previous quiz solution.
But I do not get why argf(x) changes -2$\pi$
I tried to analyze the change sign for both Im(f) and Re(f), This is what I got:
x<-1, Im(f) > 0
-1<x<0, Im(f) <0
0<x<1, Im(f) >0
x>1,Im(f) <0
for real part:
x<-$\frac{1}{2}$ ,Re(f) > 0
-$\frac{1}{2} $< x < $\frac{1}{2}$, Re(f) < 0
x > $\frac{1}{2}$, Re(f) > 0
 Then first, f moves from first quadrant to third quadrant  through second quadrant, then f moves back to first quadrant through second quadrant and f ends up in fourth quadrant. Therefore, I think arg(f) changes at most $-\pi$.

Can anyone help figure out which part is wrong?
Thank you!


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