/sci/ - Science & Math

>>11801104
The only way to properly define infinity is in the context of a single linear fractional transformation on the Riemann sphere.
If we consider the canonical map f(z)=1/z, then we define infinity to be the image of f at 0, so as to make f biholomorphic on the Riemann sphere. Or in your words, u = 1/0. Similarly, u is just the preimage of 0 by f.
u is only well-defined in the domain of f and f-inverse, you can't just yolo "u*0", it's ill-defined.

The wikipedia article is quite good and got me through the start of a grad class on complex variables.
https://en.wikipedia.org/wiki/M%C3%B6bius_transformation