Bilinear Transformation problem

[RJLiberator]

Homework Statement

Find the bilinear transformation that maps the points z1=infinity, z2=i, z3=0 to the points w1=0, w2=i, w3=infinity.

Homework Equations

w=(az+b)/(cz+d)

The answer is -1/z

The Attempt at a Solution

We have:
infinity --> 0
i --> i
0 --> infinity

Since 0 goes to infinity, it means the denominator "is 0", so therefore d must be 0 in our relevant equation right off the bat.

If we take the limit as z--> infinity, we are supposed to get w=0, so 0=a/c in the limit and we see that a=0.

So now we are left with b/cz
and i-->

i=b/(ci) and we see that -c=b

But how do I prove here that c=1, and b=-1 instead of something else?

thanks.

 

[Orodruin]

But how do I prove here that c=1, and b=-1 instead of something else?

This is irrelevant. If you take c = 2 and b = -2, you end up with the same transformation (i.e., the constants are only well defined up to an overall multiplicative factor).

 

[RJLiberator]

Hm. Excellent, so what you are saying is it does not need to be proved since it's irrelevant.
It may as well be c=100, and b=-100 and then w'ed have
-100/(100z) which reduces to -1/z

Thanks.