# Toronto Math Forum

## MAT334--2020S => MAT334--Lectures & Home Assignments => Chapter 3 => Topic started by: Yan Zhou on March 15, 2020, 11:28:52 PM

Title: 3.1 question 8
Post by: Yan Zhou on March 15, 2020, 11:28:52 PM
$$2z^4 - 2iz^3 + z^2 + 2iz -1$$
find roots for upper half line.
http://forum.math.toronto.edu/index.php?topic=1591.0
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!