[SOLVED] Conformal Mapping

Quote:

Originally Posted by ComplexXavier

What is a conformal mapping from the upper half unit disk to the entire unit disk?

The way to do questions like this is always to build up the conformal map as a composition of simpler maps. In this case, you probably already know the map $u = -\dfrac{z+1}{z-1}$ , which takes the unit disk to the right-hand half plane. You can check that this map takes the upper half unit disk to the positive quadrant. Start with that map, follow it with the map $v=u^2$ , which takes the positive quadrant to the upper half plane. Then the map $w = \dfrac{v-i}{v+i}$ takes the upper half plane to the unit disk.

Quote:

Originally Posted by ComplexXavier

What is a conformal mapping from the 3/4 plane where the argument is from 0 to 3pi/2, onto an infinite strip.

Use similar ideas for this one. You may need the map $z\mapsto z^{4/3}$ , which takes the 3/4 plane to the whole plane slit along the positive real axis; and the logarithm function, which takes the split plane to a strip.