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Homework Help: Complex mapping question

  1. Apr 17, 2012 #1
    1. The problem statement, all variables and given/known data
    find linear fractional transformation from D={z:|Arg z| < [itex]\alpha[/itex]}, [itex]\alpha≤[/itex][itex]\pi[/itex] to the upper half plane

    2. Relevant equations

    3. The attempt at a solution

    The problem I am having here what exactly D is.. (visualizing it) D is just z such that |Arg z|≤[itex]\pi[/itex] right? so wouldn't that just be the entire complex plane? If we consider the Argument from 0 to [itex]\pi[/itex] and from 0 to -[itex]\pi[/itex] since |-[itex]\pi[/itex]|=[itex]\pi[/itex] Is this correct??
  2. jcsd
  3. Apr 17, 2012 #2
  4. Apr 17, 2012 #3
    I believe what this is saying is that you first select [itex]\alpha \leq \pi[/itex] and then form [itex]D := \left\{ z : \left| \mathit{arg} \ z \right| < \alpha \right\}[/itex]. This would not be the entire complex plane. I believe for say [itex]\alpha = \frac{\pi}{2}[/itex] would look like [itex]D = \left\{ z : \mathit{real} \ z > 0 \right\}[/itex].
  5. Apr 17, 2012 #4
    right but then if [itex]\alpha[/itex]=[itex]\pi[/itex] wouldn't it just be the entire plane? since if it is [itex]\pi[/itex]/2 then it is half of the plane..
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