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Complex Mapping

  • #1
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Let [itex]Q:=\{ z: Re(z)>0, Im(z)>0, |z|<1 \}[/itex] i.e. the quarter disc.

what is the image of Q by the mapping [itex]f(z)=z^2[/itex]

by trial and error with various points, my answer is that it takes Q to the semicircle [itex]\{ z: Re(z)>0, |z|<1 \}[/itex]

but can't how this explicitly as it's not a mobius transformation with which im used to dealing with.
 

Answers and Replies

  • #2
537
3
Let [itex]Q:=\{ z: Re(z)>0, Im(z)>0, |z|<1 \}[/itex] i.e. the quarter disc.

what is the image of Q by the mapping [itex]f(z)=z^2[/itex]
The best way to find the image of mappings like this is to let [itex]z=re^{i\theta}[/itex] and then look at what your mapping does to this polar representation of all z in your region Q.
 
  • #3
1,444
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ok so the modulus will square but thats just 1 again and the argument doubles.

our original angle was from 0 to pi/2
so now we go from 0 to pi
so it will be a semicircle in the upper half plane of radius 1?
 
  • #4
537
3
so it will be a semicircle in the upper half plane of radius 1?
Looks good to me.
 

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