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
There seems to be an error in the definition of the annulus, since the lower and upper bounds for z should be different.
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