Describe a conformal map from \[ \{z:|z-4i|<4\}\cap\{z:|z-i|>1\} \] onto the open unit disk.
Solution
Compose
\[ f _ { 1 } : z \to 1 / z \]
\[ f _ { 2 } : z \to 8 \pi ( z + i / 2 ) / 3 \]
\[ f _ { 3 } : z \to e x p ( z ) \]
\[ f _ { 4 } : z \to ( z - i ) / ( z + i ) \]