Complex Analysis Mapping With Principal Branch

  • Thread starter terib
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1. Verify that f(z) = Sqrt(z^2 - 1) maps the upper half plane Im z > 0 onto the upper half plane Im w > 0 slit along the segment from 0 to i. [Hint: use the principal branch]

2. Homework Equations
We studied factional linear transformations with T(z) = (a z + b) / (c z + d) , but I can't seem to apply that here.

We defined the principal branch in the beginning of the class as -Pi < arg z < Pi

We have not discussed the slit along the segment from 0 to i, but I'm guessing the it is like a period strip from Im w = 0 to Im w = i

3. The Attempt at a Solution
I let z = x + iy, but don't know how to work with the sqrt root to get something I can work with.
 

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