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Find the mapping $w(z)$ which maps upper half plane $\Im z>0$ onto unit circle $|w|<1$ such that

$w(i)=0$ and $\arg w'(i)=-\frac{\pi}{2}$.

In general I can the find the mapping which satisfies above condition. But what does mean $\arg w'(i)=-\frac{\pi}{2}$?

I would be very grateful if anyone can explain the meaning of this condition in concrete example?

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The number $w'(i)$ is a complex number. The idea is to choose $w$ so that$$w'(i)=r\left(\cos\left(-\frac\pi2\right)+\sin\left(-\frac\pi2\right)i\right)\left(=-ri\right),$$for some $r\in(0,\infty)$.

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    $\begingroup$ Thanks a lot for answer! I guess that I got you! +1 $\endgroup$
    – RFZ
    Commented Feb 13, 2019 at 14:57

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