1. Not finding help here? Sign up for a free 30min 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!

Find a linear fractional transformation that carries circle to a line.

  1. Dec 2, 2008 #1
    1. The problem statement, all variables and given/known data

    Define L: |z| = 1 -----> Re( (1 + w)) = 0. Find L.

    2. Relevant equations

    A transformation is defined by three unique points by T(z) = (z-z1)(z2-z3) / (z-z3)(z2-z1). If we have two transformations T and S, and we want T = S for three distinct points, then we have the transformation L by the transformation S^(-1)[T(z)].

    3. The attempt at a solution

    I chose the points on the circle 1, i, and -1 to go to the points infinity, 0 and 1+i respectively. My calculations gave me L(z) = (1+i)((z-i) - (z+1))(infinity) / ((infinity)(z-1)i - (z+1)(1+i)). The book gives me u(1-i)(z+1)/(z-1) where u is any real number.

    What should I do to get this simple form (aka the right answer). Thank you.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 2, 2008 #2
    Why are you mapping points to 0 and 1 + i? Doesn't Re(1 + w) = 0 represent a vertical line?

    Cancel out the terms containing infinity.
     
  4. Dec 3, 2008 #3
    I'm sorry, that is a typo. I meant to write Re((1 + i)w) = 0. This is the line y = x. I choose two points on this line, and a point at infinity.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?