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Bilinear Transformation problem

  1. Aug 4, 2015 #1

    RJLiberator

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    1. The problem statement, all variables and given/known data
    Find the bilinear transformation that maps the points z1=infinity, z2=i, z3=0 to the points w1=0, w2=i, w3=infinity.

    2. Relevant equations
    w=(az+b)/(cz+d)

    The answer is -1/z

    3. The attempt at a solution

    We have:
    infinity --> 0
    i --> i
    0 --> infinity

    Since 0 goes to infinity, it means the denominator "is 0", so therefore d must be 0 in our relevant equation right off the bat.

    If we take the limit as z--> infinity, we are supposed to get w=0, so 0=a/c in the limit and we see that a=0.

    So now we are left with b/cz
    and i-->

    i=b/(ci) and we see that -c=b

    But how do I prove here that c=1, and b=-1 instead of something else?

    thanks.
     
  2. jcsd
  3. Aug 4, 2015 #2

    Orodruin

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    This is irrelevant. If you take c = 2 and b = -2, you end up with the same transformation (i.e., the constants are only well defined up to an overall multiplicative factor).
     
  4. Aug 4, 2015 #3

    RJLiberator

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    Hm. Excellent, so what you are saying is it does not need to be proved since it's irrelevant.
    It may as well be c=100, and b=-100 and then w'ed have
    -100/(100z) which reduces to -1/z

    Thanks.
     
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