# Bilinear map

1. Sep 29, 2011

### lolproe

1. The problem statement, all variables and given/known data
find the linear fractional transformations (bilinear transformations) which map the ponts:

$$z_{1} = -1, z_{2} = 0, z_{3} = -1$$ into $$w_{1} = j, w_{2} = \infty, w_{3} = 1$$

2. Relevant equations
N/A
3. The attempt at a solution
I really don't have anything. For every question of this type I've tried before, I used the shortcut of setting
$$\frac{(w - w_{1})}{(w - w_{2})} \frac{(w_{3} - w_{2})}{(w_{3} - w_{1})} = \frac{(z - z_{1})}{(z - z_{2})} \frac{(z_{3} - z_{2})}{(z_{3} - z_{1})}$$
So the equation goes to zero (or one) when w or z are equal to one of the given points. I just don't know how to translate that into this example. I can't just throw infinity into the equation as a constant so I really don't know what to do with it. All I can figure out is that I suppose my final expression for w will probably end up with the denominator as a multiple of z alone, seeing as 0 maps to infinity. But other than that I have no idea how to start this.

My notes for this class don't have any examples where there is a point at infinity, and there aren't any examples where the above formula can't be used. They mention that you can brute-force for all the coefficients in your final expression in w, but that's all the information given about that method so I don't know if that would work.

Thanks for any help

Just for fun, I ended up throwing infinity in as w2 in the equation and everything seemed to work out. I ended up with $w = \frac{(1+j)z + (j-1)}{2z}$ and that seems to map all the points properly, at least if $\frac{j - 1}{0} = \infty$. My main problem with this is that I had to algebraically say that $\frac{-\infty}{-\infty} = 1$ when I first subbed it in, and something about that rubs me the wrong way. The whole substituting infinity thing in general seems wrong to me, but saying that $(1 - \infty) = (w - \infty) = -\infty$ just seems a bit sloppy to me and I'm not sure if I can say that.

Last edited: Sep 29, 2011