# Thread: Complex Analysis Linear Fractional Transformation

1. ## Complex Analysis Linear Fractional Transformation

I am stuck on a LFT question,
T maps the real axis onto itself and the imaginary axis onto the circle $\displaystyle \mid w-\frac{\1}{2} \mid = \frac{\1}{2}$
I do not have a good grasp on LFTs, although I understand fixed points and triples to triples techniques and can solve problems like,
Find a LFT that carries the circle $\displaystyle \mid z \mid = 1$ onto the line $\displaystyle Re((1+i)w)=0$
But when I have to take into account multiple properties of transformation I get lost, I've read my text a bunch and even picked up another, not to mention the countless number of help articles I've read online. I'm not sure what to do. I've attempted to use fixed points and triples to triples techniques to no useful end. Any hints or reference material would be greatly appreciated!
Thank you!

2. ## Re: Complex Analysis Linear Fractional Transformation

So I think I solved this one,

$\displaystyle T(z)=\frac{z-\infty}{z}\frac{i}{i-\infty}$

$\displaystyle T(z)=\frac{zi-\infty i}{zi-z\infty}$

yada...yada... do the same triplet to triplet thing for the other set of points... and its inverse is

$\displaystyle S^-^1(w)= \frac{z(\frac{1}{2}+\frac{i}{2})}{z(\frac{1}{2} + \frac{i}{2})+(\frac{1}{2} - \frac{i}{2})}$

$\displaystyle S^-^1(T(z)) = \frac{\frac{zi-\infty i}{zi-z\infty}(\frac{1}{2}+\frac{i}{2})}{\frac{zi-\infty i}{zi-z\infty}((\frac{1}{2} + \frac{i}{2})+(\frac{1}{2} - \frac{i}{2}))}$

$\displaystyle S^-^1(T(z)) = \frac{(zi-\infty i)(\frac{1}{2}+\frac{i}{2})}{(zi-\infty i)(\frac{1}{2} + \frac{i}{2})+(zi-z\infty)(\frac{1}{2} - \frac{i}{2})}$

take the limit...

$\displaystyle L(z)=\frac{\frac{1}{2}-\frac{i}{2}}{\frac{1}{2}-\frac{i}{2}-\frac{z}{2}+\frac{zi}{2}}$

$\displaystyle L(z)=\frac{1}{1-z}$

And it turns out that this transformation leaves the real axis alone... this was a little planing (with choosing the triples) but mostly luck... Does anyone know a better way of solving this?