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I'm trying to solve the following problem.

Find an explicit conformal mapping from the region \begin{equation} \{|z| < 1\} \backslash [0, 1) \end{equation}

onto the upper half plane $\{Im z> 0\}$

I know that the maps $F(z) = \frac{i - z}{i+z}$ and $G(z) = i\frac{1 - z}{1+z}$ map the unit disc to the upper half plane. However, in the case when the disk has a segment removed I'm not sure how to modify these mamps to give a conformal mapping.

Math_Day
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  • It will not be enough to use only linear fractional transformations like your $F,G$. Do you know some other analytic functions to use? – GEdgar Apr 01 '23 at 06:10
  • Inversion $F(z) = 1/z$ comes to mind but this maps the lower half of the disk to the negative half plane – Math_Day Apr 01 '23 at 06:15
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    For a google search use the word "slit" e.g., "unit disk with a slit on real positive axis" gives this. – Jean Marie Apr 01 '23 at 06:17

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