1. ## Conformal Mapping

Hello,

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

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

Any help with this would be greatly appreciated. Thanks!

2. 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.

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.