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Homework Help: 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.
     
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