Understanding Holomorphic Functions at Infinity: Exploring Complex Analysis

[quasar987]

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]

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

 

[quasar987]

Thx shmoe!

 

[quasar987]

Why do we never talk about continuity and differentiability at infinity for real functions?

 

[matt grime]

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 learned about them.

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

 

[benorin]

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]

Perhaps I should have said 'it has proved more useful'.

It is just useful to allow infinity as a point. what is the integral round a closed curve of a meromorphic function? the sum of the residues at the simple poles. what are simple poles? places where the function takes the value infinity with multiplicity 1. By treating infinity as a genuine point then we can start talking about things more uniformly.Any holmorphic function from Cu{infinty} to C is constant which is another way of stating that theorem that any bounded holmorphic function is constant.