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

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Homework Help Overview

The discussion revolves around determining the range of the complex function p(z) = -2z^3, specifically for z values within the quarter disk defined by |z| < 1 and 0 < Arg(z) < π/2. Participants are examining the implications of this function's behavior in the specified domain.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants are questioning the validity of the proposed answer regarding the range, particularly the claim that |w| < -2 and the argument range. There is a suggestion to express z in polar form as z = re^(iθ) to analyze the function further.

Discussion Status

There is an ongoing examination of the assumptions made in the original problem statement. Some participants express skepticism about the provided answer and are collaboratively exploring alternative interpretations and calculations. The discussion is active, with participants seeking clarity on the function's behavior.

Contextual Notes

Participants note confusion stemming from the answer provided in the textbook, which they believe may be incorrect. This has prompted a deeper investigation into the function's range and the assumptions underlying the problem.

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

The answer is the circular sector [itex]|w| < -2[/itex], [itex]-\pi <[/itex]Arg[itex]w < \frac{\pi}{2}[/itex]

What's a good way of seeing why this is true?
 
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jdinatale said:
Describe the range of [itex]p(z) = -2z^3[/itex] for [itex]z[/itex] in the quarter disk [itex]|z| < 1, 0 <[/itex]Arg[itex]z < \frac{\pi}{2}[/itex].

The answer is the circular sector [itex]|w| < -2[/itex], [itex]-\pi <[/itex]Arg[itex]w < \frac{\pi}{2}[/itex]

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.
 
LCKurtz said:
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 said:
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 said:
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?
 

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