Whats a way to determine the range of this complex function?

[jdinatale]

Describe the range of $p(z) = -2z^3$ for $z$ in the quarter disk $|z| < 1, 0 <$Arg$z < \frac{\pi}{2}$.

The answer is the circular sector $|w| < -2$, $-\pi <$Arg$w < \frac{\pi}{2}$

What's a good way of seeing why this is true?

 

[LCKurtz]

Describe the range of $p(z) = -2z^3$ for $z$ in the quarter disk $|z| < 1, 0 <$Arg$z < \frac{\pi}{2}$.

The answer is the circular sector $|w| < -2$, $-\pi <$Arg$w < \frac{\pi}{2}$

What's a good way of seeing why this is true?

You certainly aren't going to get $|w|<-2$ and that argument range isn't correct either. Write $z = re^{i\theta}$ and work with that.

 

[jdinatale]

You certainly aren't going to get $|w|<-2$ and that argument range isn't correct either. Write $z = re^{i\theta}$ and work with that.

I totally agree with you. The answer in the back of the book (which is what I posted in the OP) seems to be incorrect. That's part of the reason why I was confused.

 

[LCKurtz]

You certainly aren't going to get $|w|<-2$ and that argument range isn't correct either. Write $z = re^{i\theta}$ and work with that.

I totally agree with you. The answer in the back of the book (which is what I posted in the OP) seems to be incorrect. That's part of the reason why I was confused.

So what do you get?