Hi, everyone:(adsbygoogle = window.adsbygoogle || []).push({});

I am trying to see if it is true that if we are given f:C-->C analytic ; C complex plane,

then, to extend f to a function defined on C^ (Riemann Sphere) , i.e, to get:

f^: C^ -->C^

with f^|_C =f

i.e., the restriction of f^ to the complex plane agrees with f ,

If we need to define f(oo) =oo .

I think the answer is yes. Here's what I have:

We consider a 'hood ('hood:=neighborhood.) W of oo in C^ , which are complements

of compact 'hoods K in C , together with {oo}, i.e., W=C\K U {oo} , for K compact in C.

By Liouville's thm., |f|-->oo on balls B(0;r) , as r-->oo . And then by continuity,

it would seem that we need f(oo)=oo, since W= C-B(0;r) is a 'hood of oo.

Alternatively, if we had an analytic map f on C , f would go to oo on balls B(0;r)

as r->oo . Then if we used the stereo projection S: C-->C^ , and push f

forward by this projection, we would have Sof (composition)-->oo .

But this is still not rigorous-enough.

Any Ideas?.

Thanks.

P.S: If it bothers people to use regular ASCII, please let me know. I use ASCII

as a way to force myself to keep things clear . But I can change if neccessary.

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Extending Analytic Functions f:C->C , to f^:C^->C^; C^=Riemann Sphere

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

**Physics Forums - The Fusion of Science and Community**