Can I think of the conformal mapping w = (z+1/z) as a linear fractional transformation? The mapping 
$$w = z + \frac{1}{z}$$
looks linear in $z$.  However, it would not be in the form 
$$\frac{Az+B}{Cz+D}$$
since putting the two terms together gives 
$$\frac{z^2+1}{z}$$
So my question is:  is this mapping a linear fractional transformation?
I am hoping it is, so that I can determine its action on a half disk, using the nice and familiar symmetry properties of LFTs.
If it is not an LFT, are there symmetry properties of this mapping to notice?
One specific nice property of LFTs that I have in mind is that symmetric points w.r.t. to a circle, under an LFT, are again symmetric points w.r.t. the image of that circle.
Thanks,

No: linear fractional transformations are bijective, and this map isn't: consider $z=2$ and $z=1/2$.
You can take a look at the graph here: http://davidbau.com/conformal/#z%2B1%2Fz
There is some nice symmetry in the fact that $f(z)=f(1/z)$, so it's "symmetric about the unit circle" (up to flipping across the real axis), in some sense. There are probably other nice properties, but this is the first that comes to mind.
