Extending complex functions f:C->C into f^:C^->C^

In summary: Sorry for the long answer, but hopefully it is of some help.In summary, the function f: ℂ-->ℂ' must be a proper function in order to be extended into f^:C^-->C^, and the only meromorphic functions are the rational functions.
  • #1

Bacle2

Science Advisor
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Extending complex functions f:C-->C into f^:C^-->C^

Hi, All:

I have a function f: ℂ-->ℂ' , i.e., a complex-valued function (use ℂ,ℂ' to make a distinction between the complexes as domain and codomain respectively), and i want to extend it into
the Riemann Sphere ℂ^, i.e., I am looking for f^ such that f^|=f. If I remember correctly, a necessary and sufficient condition for extensibility is
that f must be a proper function, i.e., that for every K compact in ℂ' is sent to a compact
set in ℂ, i.e., f-1(K) is compact in ℂ? I think this somehow had to see with
ℂ^ being the 1-pt compactification of ℂ; is this correct? If so, anyone have a ref. for the
proof, if not, could someone please let me know what the correct result is?

Thanks.
 
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  • #2


Indeed, the topology on the Riemann sphere is

[tex]\{U~\vert~U\cap\mathbb{C}~\text{is open in}~\mathbb{C}~\text{and if}~\infty\in U~\text{then}~U^c~\text{is compact}\}[/tex]

Proving that the inverse of an open set is open under a proper map is now very easily done.
 
  • #3


Thanks, Micromass, but I was hoping for a holomorphic extension. I was thinking of using some form of Riemann Removable singularity near ∞ , to avoid having to use manifold charts, or some heavy-handed methods. I know the functions that extend continuously in the real case are those that go to ∞ as x→∞ itself, but I don't see clearly how to extend holomorphically into C^.
 
  • #4


Ah, so you want to extend a holomorphic function to a meromorphic function??

Well, then I need to say that the only meromorphic functions [itex]\hat{f}:\hat{\mathbb{C}}\rightarrow \hat{\mathbb{C}}[/itex] are the rational functions. So the condition you need to put on f is that it is a rational function.
 
  • #5


I think rational functions f: C^-->C^ are holomorphic on C^ --tho I am not clear on how we define holomorphic on maps C^-->C^ without manifold charts of some sort, or maybe some function fields , since ∞ is not a problem
value anymore in C^. If we have, e.g.,az+b in the denominator, then z=-b/a is not
a singularity anymore. Maybe this is an issue for Algebraic Geometry.
 
  • #6


Never mind, Micromass, I got it; thanks.
 
  • #7


Sorry, just in case anyone else reads and is interested in the same issue: outside of ℝn we cannot just differentiate; in manifolds,we need to pull back functions by chart maps and _then_ differentiate. In our case of the Riemann sphere, which is an analytic variety (i.e., a complex manifold), we use the stereographic projection , using the fact that the transition map in the stereographic projection is given by 1/z , we need to pre- and post- compose with 1/z (which will take us into ℂ) , where we can use the standard notion of differentiability. This way we can decide when a function is analytic in the Riemann sphere.
 

1. What does it mean to extend a complex function?

Extending a complex function f:C->C into f^:C^->C^ means to find a way to define the function for input values that were previously undefined. This allows the function to be used for a larger set of input values.

2. Why is it important to extend complex functions?

Extending complex functions allows us to work with a larger set of input values and can help us better understand the behavior of the function. It also allows us to solve more complex problems that involve these functions.

3. How do you extend a complex function?

The process of extending a complex function involves analyzing the behavior of the function and finding ways to define it for input values that were previously undefined. This can be done through methods such as analytic continuation or using properties of the function.

4. Are there any limitations to extending complex functions?

There may be limitations to extending complex functions, as some functions may not have a unique extension or may have a singularity at certain points that cannot be extended. Additionally, extending a function may change its behavior in unexpected ways.

5. How does extending complex functions relate to the study of complex analysis?

Extending complex functions is an important concept in the field of complex analysis, as it allows us to explore the behavior of functions in a larger context. It also helps us understand the properties of complex functions and how they can be used to solve problems in mathematics and science.

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