1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Bilinear map

  1. Sep 29, 2011 #1
    1. The problem statement, all variables and given/known data
    find the linear fractional transformations (bilinear transformations) which map the ponts:

    [tex]z_{1} = -1, z_{2} = 0, z_{3} = -1[/tex] into [tex]w_{1} = j, w_{2} = \infty, w_{3} = 1[/tex]

    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
    [tex]\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})}[/tex]
    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

    [edit]Just for fun, I ended up throwing infinity in as w2 in the equation and everything seemed to work out. I ended up with [itex]w = \frac{(1+j)z + (j-1)}{2z}[/itex] and that seems to map all the points properly, at least if [itex]\frac{j - 1}{0} = \infty[/itex]. My main problem with this is that I had to algebraically say that [itex]\frac{-\infty}{-\infty} = 1[/itex] 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 [itex](1 - \infty) = (w - \infty) = -\infty[/itex] just seems a bit sloppy to me and I'm not sure if I can say that.
     
    Last edited: Sep 29, 2011
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Bilinear map
  1. Linear Maps (Replies: 0)

  2. Polynomial Mapping (Replies: 0)

  3. Conformal mapping (Replies: 0)

  4. Conformal mapping (Replies: 0)

Loading...