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

  • Thread starter 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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  • #2
LCKurtz
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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?
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.
 
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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.
 
  • #4
LCKurtz
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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?
 

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