Map complex line to complex circle

[bmanbs2]

Homework Statement

Find the Linear Fractional Transformation that maps the line Re\left(z\right) = \frac{1}{2} to the circle |w-4i| = 4.

Homework Equations

For a transform L\left(z\right),

T\left(z\right)=\frac{z-z_{1}}{z-z_{3}}\frac{z_{2}-z_{3}}{z_{2}-z_{1}}

S\left(w\right)=\frac{w-w_{1}}{w-w_{3}}\frac{w_{2}-w_{3}}{w_{2}-w_{1}}

For S\left(w\right) = \frac{aw+b}{cw+d}

S^{-1} = \frac{-dw+b}{cz-a}

And the final transform is L\left(z\right) = S^{-1}\left(T\left(z\right)\right)

The Attempt at a Solution

I know how to calculate the transform for any three points to any other three points, so may I just pick any three points on the line and the circle? If not, how do I pick the correct three points?

 

[lanedance]

not the only way to do it but i would break it into the following steps for clarity

- start with the line Re(z) = 1/2, given by z(y) = 1/2+iy
- shift the line to the imaginary axis Re(z)=0
- map the line to the interval [0,2pi), this part is the key step
- map [0,pi) onto the unit circle
- scale and shift the circle to its given centre and radius