Mobius Transformations-Complex Analysis

[WannaBe22]

Homework Statement

Mobius Transformation copies the annulus $$\{ z: r<|z|<1 \}$$ to the region bounded by the discs $$\{ z : |z-\frac{1}{4}| = \frac{1}{4} \}$$ and
$$\{ z: |z|=1 \}$$ .

Find r


Hope you guys will be able to help me!


Thanks a lot!

Homework Equations

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 $\alpha$ are complex numbers such that $\overline{\alpha}z \neq 1$. then

$$\left | \frac{z - \alpha}{1 - \overline{\alpha}z} \right | < 1,$$

if |z| < 1 and $|\alpha|$ < 1, and

$$\left | \frac{z - \alpha}{1 - \overline{\alpha}z} \right | = 1,$$

if |z| = 1 or $|\alpha|$ = 1.

From this, it follows that the transformation

$$T(z) = \frac{z - \alpha}{1 - \overline{\alpha}z}$$

maps the unit disk to the unit disk. Now find a value of $\alpha$ 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