
Linear Transformation
Hello,
I'm trying to find the linear fractional transformation that maps points
z1 = i, z2 = 0, z3 = i onto points w1 = 1, w2 = i, w3 = 1.
I have set it up using the form:
(ww1)(w2w3) / (ww3)(w2w1) = (zz1)(z2z3) / (zz3)(z2z1) such that in my example (w+1)(i1) / (w1)(i+1) = (z+i)(0i) / (zi)(i).
Multiplying the left side by (i1) / (i 1), I have i[(w+1) / (w1)] = the right side of the equation above.
Getting rid of the i on the left side, I have (w+1)/(w1) = i[(z+i)/(zi)].
I'm stuck at the point where I have: (w+1)(zi) = (iz  1)(w1). Is this correct, and what do the final steps look like from here?
Thank you.

I got
$\displaystyle \frac{w+1}{w1}=i \frac{z+i}{zi}=\frac{zi1}{zi}$
Expressing w
$\displaystyle w=i \frac{1z}{1+z}$