# Holomorphic at infinity

Homework Helper
Gold Member
Hi, could someone explain what it means for a function to be holomorphic on $\mathbb{C}\cup \{\infty\}$? More precisely, what does it mean for it to be holomorphic at $\infty$. Thx.

shmoe
Homework Helper
f(z) is holomorphic at infinity if f(1/z) is holomorphic at 0. Likewise for singularities.

Homework Helper
Gold Member
Thx shmoe!

Homework Helper
Gold Member
Why do we never talk about continuity and differentiability at infinity for real functions?

matt grime
Homework Helper
because infinity is not part of the real line and not in terms of analysis a useful point to add in, whereas infinity is a very useful adjunction to the complex plane: poles and zeroes are far more important to a study of complex analysis than real analysis. That's just the way it has worked out, and is a very hand wavy explanation. Sorry.

The point is, trying again, that Cu{infinity} is a very important compact riemann surface. It is useful to add this point and talk about functions which take infinity as an admissible value in complex analytic terms and not for real things.

But, of course, we do talk about such things for the extended real line, but they just aren't as useful, and therefore *you* haven't learnt about them.

(You do make a large implicit generalization from what you know about to what *we* know about).

benorin
Homework Helper
Gold Member
matt grime said:
The point is, trying again, that Cu{infinity} is a very important compact riemann surface. It is useful to add this point and talk about functions which take infinity as an admissible value in complex analytic terms and not for real things.

Is this the so-called "one point compactification" of either the complex plane or the real line? Would you also expound upon the significance of the complex characteristics of infinity (as, say, a point in the complex plane)?

TIA,
-Ben

matt grime