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Homework Help: Trouble finding a mobius transformation from a domain to a unit disc

  1. Mar 11, 2010 #1
    1. The problem statement, all variables and given/known data
    S = { z | |Im(z)| < 5 }, z is a complex number


    2. Relevant equations
    I am trying to generate a mobius transformation w = f(z) such that it will map S onto a unit disc but I keep running into problems and contradictions. I think there is a big mistake in my attempt but I can't figure it out, can somebody please give me a hint on what is wrong?


    3. The attempt at a solution
    First I know that |Im(z)| < 5 implies that -5 < y < 5 and I want to find a mobius transformation such that S will be mapped onto unit disc, | z | < or = 1.
    I start with the standard Mobius transformation form:

    f(z) = (az+b)/(cz+d), assume c doesn't equal 0, and I want the following,
    0 -> 0
    5i -> i
    infinity -> 1

    Then f(0) = b/d = 0, I choose b = 0 instead of d = infinity.
    f(infinity) = a/c = 1, this shows a = c
    f(5i) = (5ai)/(5ci + d) = (-25ac - 5adi)/((-25c^2) - d^2) = (25ac + 5adi)/(25c^2 + d^2)
    However at this point I figure out that (25c^2)/(25c^2 + d^2) = 0, this implies that c = 0.
    Since c = 0 and a = c, then a = 0.
    Now I can't figure out what d is, since I have 0/(d^2) = 1, which makes no sense.
    In addition because a = b = c = 0, ad - bc = 0 which violates the condition of Mobius transformation.
     
    Last edited: Mar 11, 2010
  2. jcsd
  3. Mar 11, 2010 #2
    There's your first problem.
     
  4. Mar 11, 2010 #3
    Sorry I forgot to add the mod, it should be |Im(z)|<5
     
  5. Mar 11, 2010 #4
    Mobius transformations take generalized circles to generalized circles. The boundary of the unit disk is a circle. What's the boundary of the set you're starting with? Is it a (generalized) circle?
     
  6. Mar 11, 2010 #5
    In this case, I think my S is not a generalized circle so I can't generate a mobius transformation?

    If not, can I still find an analytic functions that maps S onto unit disc?
     
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