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Complex analysis question

  1. Oct 1, 2013 #1
    1. The problem statement, all variables and given/known data


    Let S = {z : 1<= Im(z) <=2}. Determine f(S) if f: S ->C
    defined by
    f(z) = (z + 1) / (z - 1)





    2. Relevant equations

    z = x + iy

    3. The attempt at a solution
    [attempt at solution]

    so here my solution

    f(z) = 1 + 2/(z - 1)

    after doing some algebra <-> f(z) = x^2 + y^2/((x - 1)^2 + y^2) - [2y/((x-1)^2 + y^2)]i

    therefore Im(z) = -2y/((x - 1)^2 + y^2) so F(S) = {z : 1<= (-2y)/((x-1)^2 + y^2) <= 2}
    but I am stuck at this point I don't know wat does this represent.
     
  2. jcsd
  3. Oct 1, 2013 #2

    Dick

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    Homework Helper

    Concentrate on what the boundaries of your region are. For example, if 1=(-2y)/((x-1)^2 + y^2) what kind of curve is that? Multiply it out and complete the square. At a more abstract level f(z) is a Mobius transformation. It will map lines to lines or circles, yes?
     
    Last edited: Oct 1, 2013
  4. Oct 2, 2013 #3
    yes I did that I got something weird

    I got (x-1)^2 + y^2 <= -2y <= 2( (x - 1)^2 + y^2)) the way I see it its between two circles but how to show that ???
     
  5. Oct 2, 2013 #4

    Dick

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    Just look at the boundaries. Where your inequality becomes an equality. 1=(-2y)/((x-1)^2 + y^2) and 2=(-2y)/((x-1)^2 + y^2). What are the boundary curves? And yes, they are two circles.
     
  6. Oct 2, 2013 #5
    o I see I figured it out ty alot Dick!
     
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