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Math Help - mobius transformation question

  1. #1
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    mobius transformation question

    they said that
    z=-1 -> w=infinity
    z=0 -> w=i
    z=1 -> w=0

    and that that
    z=i -> w=(1+i)/(1+i)

    how the got that the last point should go to (1+i)/(1+i)

    and after they know which point goes to what point
    on what formula they input the point to find the trasformation formala??
    how they deal with the point where the need to put infinity
    ??
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  2. #2
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    There's a theorem (on p. 64 of Gamelin, for example) that says the following:

    Given any three distinct points z_{0}, z_{1}, z_{2} in the extended complex plane, and given any three distinct values w_{0}, w_{1}, w_{2} in the extended complex plane, there is a unique fractional linear transformation [Ackbeet: Mobius transformation] w=w(z) such that w(z_{0})=w_{0}, w(z_{1})=w_{1}, and w(z_{2})=w_{2}.

    So, applying this to your situation, the Mobius transformation is completely determined by the equations:

    z=-1 -> w=infinity
    z=0 -> w=i
    z=1 -> w=0

    My guess is that, once you've found that transformation, you'll find that i maps to the number they say. Does that help?
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  3. #3
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    can ou be more specific
    what formula to use
    ?
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  4. #4
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    Try a formula like

    w(z)=a\,\dfrac{z-b}{z-c}.

    You can fill in a, b, and c by looking at where things need to map.
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  5. #5
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    but if i put infinity instead of the w
    what to do in this case
    its not a linear equation anymore

    i couldnt solve a system of equation with infinity in one place
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  6. #6
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    What z-value gets mapped to infinity?
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  7. #7
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  8. #8
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    So what do you suppose c should be?
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  9. #9
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    so the infinity point is like a gift
    it tells us straight away one of the parameters
    without soling theequation thanks
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  10. #10
    A Plied Mathematician
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    Exactly. There might be another gift. Can you see it?
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