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How can i find an example of a

  • Bijective conformal map between $$S= \{x+iy: 0 < x < 1, 0 onto $\mathfrak{H} = \{re^{i\theta}: 0

I know conformal maps preserves angles. Any insight of how to deal with such question would be helpful.

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    Unfortunately, such a map cannot be expressed by a simple formula. It will involve the inverse of an incomplete elliptic integral of the first kind. This integral is given explicitly in the Wikipedia article linked in the answer below.2012-09-26

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