# Fractional Linear transformation from the right half plane to unit disk

I am trying to find a fractional linear transformation from the right half plane $\{ z : Re(z) >0\}$ to the unit disk. In this kind of question, I have little idea of where to start. What I usually try to do, is considering the general form of a fractional linear transformation $f(z) = \frac{az+c}{cz+d}$, I try finding the adequate $a,b,c,d$ such that the maps from the right half plane are sent to the points I want. For instance, in this case I tried sending $i \rightarrow 1$, $0 \rightarrow i$ and $-i \rightarrow -1$. However, I suspect there should be a more direct approach. Could anyone suggest or give a hint for a more efficient way to find these kind of maps?

• Look up the cross ratio. It's a specific notation/formula to do what you want – Mark May 9 '16 at 18:03
• Hint: The right halfplane are exactly those complex numbers which are closer to $+1$ than to $-1$. – Martin R May 9 '16 at 18:36