# Complex analysis/linear fractional transformation

 Sci Advisor HW Helper P: 2,020 A linear fractional transformation (lft) takes circles to circles. Since the lft under consideration here is assumed to be an automorphism of the unit disk, it must take points inside the disk to points inside the disk (i.e. inversion in any circle inside the disk is ruled out), so it must take the unit circle to itself. Alternatively, you can go the long route: starting from the formula $$z\mapsto f(z) = \frac{az+b}{cz+d}$$ (with ##ad-bc=1##, wlog), show that the stipulation ##|z|\leq1 \implies |f(z)|\leq1## puts some severe restrictions on what a,b,c,d could be. And then conclude that points with ##|z|=1## get mapped to points with ##|f(z)|=1##. This will be fairly messy though.