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Math Help - Conformal Mapping

  1. #1
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    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!
    Last edited by ComplexXavier; November 25th 2010 at 08:39 PM. Reason: User Error
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  2. #2
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    Quote Originally Posted by ComplexXavier View Post
    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 View Post
    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.
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