# Map complex line to complex circle

## 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?

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lanedance
Homework Helper
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