Conformal Mapping (1)

[Camillo]

Find the conformal mapping that transforms the circle $ |z | < 1 $ into the halfplane : $Im z > 0 $ .

[elgiovo]

Well Camillo, I think there are many conformal mappings sending the unit circle to the upper half-plane.
Let $e^(i theta)$, $e^(i phi)$, $e^(i psi)$ be three complex numbers lying on the unit circle, with $0<=theta Obviously, we need a Mobius transformation. It's well-known that there exists a unical Mobius transformation sending three points $q,r,s$ to other three points $hat q, hat r, hat s$, and it can be calculated by solving this equation:
$((w-hat q)(hat r - hats))/((w-hat s)(hat r -hat q))=((z- q)( r -s))/((z- s)( r - q))$, where $w=M(z)$.
Now, we must also remember that if $q,r,s$ are sent by $M$ respectively to $1, oo $ and $0$ and they induce a positive orientation on a circle $C$ then the interior of $C$ is mapped to the upper-half plane. Well, this was the crux move, since we are now able to calculate the desired $M$. A simple calculation yields $M(z)=((z-e^(i theta))(e^(i phi) - e^(i psi)))/((z-e^(iphi))(e^(itheta)-e^(ipsi)))$. Of course, as said before, we need that $0<=theta

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[SonjaKovaleskaja]

suggestion: it's a well known result obtained by a linear fractional function...