 Solution

A conformal automorphism of the upper half-plane is a real Möbius transformationwith coefficients determined up to a common nonzero factor. The boundary interval $(\infty ,1)$ is the oriented real boundary arc from infinity to $1$, namely the finite real interval $(-\infty ,1)$. The map must take the left endpoint $-1$ to infinity and the right endpoint infinity to $1$, while fixing zero. These three boundary values determine the Möbius transformation uniquely:Its determinant is positive, $\mathrm{Im}\varphi (z)=\mathrm{Im}z/|1+z{|}^2>0$, and on $(-1,\infty )$ its derivative is $(1+x{)}^{-2}>0$. Its endpoint limits are $-\infty$ at $-1+$ and $1$ at $+\infty$, so it has exactly the required boundary extension.