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Complex Mapping

  1. Apr 18, 2009 #1
    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.
     
  2. jcsd
  3. Apr 18, 2009 #2
    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.
     
  4. Apr 19, 2009 #3
    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?
     
  5. Apr 19, 2009 #4
    Looks good to me.
     
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