# linear fractional transformation of the complex plane that goes to the unit disc

Find all linear fractional transformation that maps {$z:Im(z)>0$} to {$w:|w|<1$}

Hint: Find one, call it $T$. Suppose there is another. Call it $S$. Then $TS^{-1}$ maps the disc conformally to itself, hence has a very precise form. Use your knowledge of this form and the inverse of $T$ to find a general form for $S$.