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Question:

Suppose that $f(z)$ is a conformal mapping of $\{\mathrm{Im}(z)>0\}$ onto the unit square $$\{0<\mathrm{Re}(z)<1,\;0<\mathrm{Im}(z)<1\}$$ such that the boundary points $0$, $1$, $\infty$ correspond to $0$, $1$, $1+i$ respectively.

a) Which point $x$ on the real axis corresponds to the vertical $i$?
b) Prove that $f^{-1}$ has an analytic continuation to a meromorphic function $F$.

Thanks.

I think it should be related to Schwartz Christoffel.

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    Do you mean the unit square :$$ \mid Re(z) \mid < 1 \space and \space \mid Re(z) \mid <1 ?$$2012-12-29

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