I'm trying to teach myself complex analysis, and am reading about linear fractional transformations.
I want to find the transformation carrying the circle $|z|=2$ into $|z+1|=1$, $-2$ into the origin, and the origin into $i$.
My text states that the transformation is determined by prescribing a point $z_1$ on circle $C$ to correspond to a point $w_1$ on C', and a point $z_2$ not on $C$ to correspond to $w_2$ not on C'. Then $z^\ast_2$ corresponds to $w^\ast_2$, the symmetric points of $z_2$ and $w_2$ in $C$ and C', respectively. Then the transformation is obtained from the relation $(w,w_1,w_2,w^\ast_2)=(z,z_1,z_2,z^\ast_2)$ (those are cross-ratios).
I let $C$ be the circle $|z|=1$, and C' be $|z+1|=1$. Then by the notions above $z_1=-2$, $w_1=0$, $z_2=0$, $w_2=i$. I calculate $z^\ast_2=\infty$ and $w^\ast_2=\frac{-1+i}{2}$.
So following what I've read, I have $(w,0,i,(-1+i)/2)=(z,-2,0,\infty)$. I don't understand how to gather the explicit form of the desired linear fractional transformation is from this, since $w$ and $z$ seem unknown to me. Could someone please explain what the transformation is now? Thank you.