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I've studied basic conformal mappings in a complex analysis class I took (mostly with discs and rectangular regions), but I've recently been faced with finding other conformal mappings that I've found my knowledge has left me....unfit.... to solve.

Finding bijective conformal mappings from

a) between the branches of the hyperbola $xy=1$ to the upper half plane

b) under the parabola $y=x^2$ to upper half plane

I've been trying to construct maps to discs (something I'm more familiar with) and then take it to the upper half plane (which is easy), but I am not really making any progress.

Thoughts or Hints?

Much appreciated!

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