Mobius Transformations-Complex Analysis

WannaBe22

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

Petek

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

WannaBe22

You're right... I'm sry... I've corrected my typo

Petek

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