- #1

quasar987

Science Advisor

Homework Helper

Gold Member

- 4,783

- 18

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter quasar987
- Start date

- #1

quasar987

Science Advisor

Homework Helper

Gold Member

- 4,783

- 18

- #2

shmoe

Science Advisor

Homework Helper

- 1,992

- 1

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

- #3

quasar987

Science Advisor

Homework Helper

Gold Member

- 4,783

- 18

Thx shmoe!

- #4

quasar987

Science Advisor

Homework Helper

Gold Member

- 4,783

- 18

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

- #5

matt grime

Science Advisor

Homework Helper

- 9,395

- 4

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).

- #6

- 1,307

- 108

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

- #7

matt grime

Science Advisor

Homework Helper

- 9,395

- 4

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.

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.

Last edited:

Share: