How to compute a linear fractional transformation that maps a circle to a given circle? How to compute a linear fractional  transformation that maps a circle to a given circle? For example, let $C_1$ be the circle $|z+(2+i)|=1$ and $C_2$ be the circle $|z-5|=7$. How to find a linear fractional transformation that maps $C_1$ to $C_2$? Thank you very much.

As I said in your previous question, a LFT is uniquely determined by 3 points and their images.  If $z_1, z_2, z_3$ are your inputs (pick 3 points on your first circle) and $w_1, w_2, w_3$ are your images of these (pick 3 points on your second circle).  Be sure they are in the same order as you move around the circle, as I mentioned before that LFTs preserve order.  Then, use the formula
$$\frac{(z_1 - z_3)(z_2 - z)}{(z_2 - z_3)(z_1 - z)} = \frac{(w_1 - w_3)(w_2 - w)}{(w_2 - w_3)(w_1 - w)}$$
and then solve for $w$ to get $w$ as a function of $z$.  This formula is one you need to know and is useful in many, many situations, not just this one.
