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

[StumpedPupil]

Homework Statement

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

Homework 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)].

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.

 

[mutton]

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.

 

[StumpedPupil]

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.