# Conformal Mapping

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• Nov 25th 2010, 05:44 AM
ComplexXavier
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!
• Nov 25th 2010, 11:41 AM
Opalg
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 \$\displaystyle 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 \$\displaystyle v=u^2\$, which takes the positive quadrant to the upper half plane. Then the map \$\displaystyle 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 \$\displaystyle 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.