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Math Help - conformal map, slit plane

  1. #1
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    conformal map, slit plane

    Find a conformal map of the slit plane \mathbb{C} \backslash (-\infty, 0] onto the open unit disk satisfying w(0)=i, w(-1+0i)=+1, w(-1-0i)=-1. What are the images of circles centered at 0 under the map? Sketch them.

    The back of the book says that w=\frac{-i(\sqrt{z}-1)}{\sqrt{z}+1} works. I think for the second part of the question I can consider three points on the circles to find the images. However, I don't see how they found this map. Thanks.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Hint : the map z \mapsto \frac{z-i}{z+i} maps the upper half-plane \mathbb{H} to the unit disc K, and the map z \mapsto z^{1/2} maps the slit plane to the upper half-plane, provided you pick the appropriate branch of the function.
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