Mobius Transformations-Complex Analysis

by WannaBe22
Tags: analysis, mobius
 P: 76 1. The problem statement, all variables and given/known data 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! 2. Relevant equations 3. The attempt at a solution Got no idea...Hope you'll be able to help
 P: 361 Mobius Transformations-Complex Analysis 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