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So I have a map from the unit disc to the upper half plane:

$$f(z) = \frac{z+i}{iz+1}$$

My logic is to now rotate this map by multiplying by $-i$ to arrive at the right half plane - though this does not seem to be a correct answer - what is the mistake I am making here?

Thanks

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    $\begingroup$ What do you get as the final result, and why do you think that it is not correct? $\endgroup$
    – Martin R
    Commented Jan 28, 2021 at 12:28
  • $\begingroup$ I've mapped this using an online complex mapper and the second part of the question I'm attempting uses the function derived to produce the Koebe function by dilating by 1/2 - squaring then taking away a quarter - I can't seem to derive said Koebe function this way. $\endgroup$ Commented Jan 28, 2021 at 12:31
  • $\begingroup$ And as a final result I get $\frac{-iz+1}{iz+1}$ $\endgroup$ Commented Jan 28, 2021 at 12:32
  • $\begingroup$ Your mapping to the upper half plane looks already wrong. It should be something like $f(z) = i \frac{1+z}{1-z}$. $\endgroup$
    – Martin R
    Commented Jan 28, 2021 at 12:33
  • $\begingroup$ Crikey! Thankyou for spotting this as it's in the lecture notes - I shall try your suggested function and let you know! $\endgroup$ Commented Jan 28, 2021 at 12:34

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I find it simpler to remember what the conformal mappings from a half plane to a disk are. As an example, the right half plane is the locus of points which are closer to $+1$ than to $-1$, i.e. the points $w$ for which $|w-1| < |w+1|$. It follows that $$ T(w) = \frac{w-1}{w+1} $$ is the Möbius transformation which maps the right half plane onto the unit disk. Now you can solve the equation $z = \frac{w-1}{w+1}$ for $w$ to get the inverse mapping $$ S(z) = T^{-1}(z) = \frac{1+z}{1-z} $$ which maps the unit disk onto the right half plane.

This is related to your result via $$ -i f(z) = -i \frac{z+i}{iz+1} = \frac{-iz+1}{iz+1} = S(-iz) $$ which is therefore correct as well: $-if$ is a rotation by $-i$ which maps the unit disk onto itself, followed by $S$ which maps the unit disk onto the right half plane.

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