conformal mappings of Riemann sphere

Question

We know that Möbius transformations are the conformal selfmaps of the Riemann sphere.

But what does being conformal at infinity correspond to?what does it mean?

Answer 1

A complex function $f(z)$ is conformal at infinity if the function $f(1/z)$ is conformal at 0. This equivalent to the usual definition of conformal in terms of leaving angles unchanged in size and sense (clockwise/anticlockwise) if you think of $f$ as being conformal at infinity if it leaves angles unchanged at the "North Pole" of the Riemann sphere.

For example, if you take the function $f(z)=\displaystyle\frac{1}{z+1}$, then $f(1/z) = \displaystyle\frac{1}{1/z+1} = \frac{z}{1+z}$. which is conformal at 0, so that $f(z)$ is conformal at infinity.