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Mobius Transformations-Complex Analysis

  1. Aug 13, 2010 #1
    1. The problem statement, all variables and given/known data
    Mobius Transformation copies the annulus [tex] \{ z: r<|z|<1 \}[/tex] to the region bounded by the discs [tex] \{ z : |z-\frac{1}{4}| = \frac{1}{4} \} [/tex] and
    [tex]\{ z: |z|=1 \}[/tex] .

    Find r


    Hope you guys will be able to help me!


    Thanks a lot!


    2. Relevant equations
    3. The attempt at a solution
    Got no idea...Hope you'll be able to help
     
    Last edited: Aug 13, 2010
  2. jcsd
  3. Aug 13, 2010 #2
    There seems to be an error in the definition of the annulus, since the lower and upper bounds for z should be different.
     
  4. Aug 13, 2010 #3
    You're right... I'm sry... I've corrected my typo
     
  5. Aug 14, 2010 #4
    OK, here are some hints:

    Do you know, or can you prove, that if z and [itex]\alpha[/itex] are complex numbers such that [itex]\overline{\alpha}z \neq 1[/itex]. then

    [tex]\left | \frac{z - \alpha}{1 - \overline{\alpha}z} \right | < 1,[/tex]

    if |z| < 1 and [itex]|\alpha|[/itex] < 1, and

    [tex]\left | \frac{z - \alpha}{1 - \overline{\alpha}z} \right | = 1,[/tex]

    if |z| = 1 or [itex]|\alpha|[/itex] = 1.

    From this, it follows that the transformation

    [tex]T(z) = \frac{z - \alpha}{1 - \overline{\alpha}z}[/tex]

    maps the unit disk to the unit disk. Now find a value of [itex]\alpha[/itex] such that T maps {z: |z| < r} to {z: |z - (1/4)| < 1/4}. That should do it.

    Please post again if you have any other questions.

    Petek
     
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