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Find Mobius Transformations (M→N)

  1. Apr 22, 2012 #1
    1. The problem statement, all variables and given/known data

    H is the upper-half plane model of the hyperbolic space

    Find all Mobius transformations that send M to N.

    2. Relevant equations

    a) M = {0, 1, ∞}, N = {∞, 0, 1}
    b) M = {0, 1, ∞}, N = {0, ∞, 2}
    c) M = {i, -i, 3i}, N = {∞, i + 1, 6}


    3. The attempt at a solution

    Using the transformation:

    5e40aaf0700c2d00dc6d5d089cba2749.png

    Could I develop two equations m(z) and n(z) that map M and N to {0, 1, ∞}, respectively? Then just find n-1[itex]\circ[/itex]m? Wouldn't that result in M→N?

    If that's true, I'm just confused as to what points to use for z1, z2, z3.
     
  2. jcsd
  3. Apr 22, 2012 #2

    Dick

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    That works. Your transformation f1(z) takes z1->0, z2->1 and z3->∞. Which case of finding a specific transformation m(z) or n(z) are you confused about? Are you supposed to map the sets in the given order or in any order?
     
    Last edited: Apr 22, 2012
  4. Apr 22, 2012 #3
    Well if I plug those z's in, I get z(-∞)/(z-∞) for m(z). I seem to hit a wall there. Same with n(z). I don't know how to fix those equations into a form that I can use for the composition.
     
  5. Apr 22, 2012 #4

    Dick

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    Which part are you working on? I suspect it's trying to map {0,1,∞} to {0,1,∞}. If z is finite then (-∞)/(z-∞) should cancel to 1. That would mean m(z)=z should work. It does, doesn't it?
     
  6. Apr 22, 2012 #5
    Yeah I see what you're saying. Then does that mean I just switch the z's around in (a) for n(z)? But use the same transformation?
     
  7. Apr 23, 2012 #6
    I cant find a transformation n such that n(1) = infinity
     
  8. Apr 23, 2012 #7

    Dick

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    Why not? Find it the same way you found m(z). What do you get when you put the z1, z2, z3 in?
     
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