Question

7. Transformations.

(1) Find a Mo ̈bius transformation that maps the open half-disk S = {z : |z| < 1, Im z > 0} to a quadrant.

(2) Find a 1 − 1 conformal mapping of the quadrant to the upper half-plane.

(3) Find a M ̈obius transformation mapping the upper half-plane to the unit disk D = {z : |z| < 1}.

(4) Compose these to give a 1 − 1 conformal map of the half-disk to the unit disk.

(5) What goes wrong if we directly use w = z2 to map S conformally and 1 − 1 onto D?

Answer 1