What is the definition of analytic at infinity in complex analysis?

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In summary: Can you help me out a bit more there?In summary, the phrase "analytic at infinity" is used to refer to a situation where a particular function or operator is analytic on all of its domain, or all of its range.
  • #1
gauss mouse
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Hi, I'm reading "Spectral Theory of Linear Operators" by John Dowson. I've seen the phrase "analytic at infinity" popping up very early in the book, but no definition is given. I wonder if anyone could tell me what the definition is or where I might find the definition and perhaps a few basic results on analyticity at infinity? The operators I'm looking at are continuous linear maps from a complex Banach space to itself, so my question is really about complex analysis.

I've tried Google but have found no definition.

I think "bounded at infinity" is a related concept, but again I do not know what that means. Maybe someone can help me out there too?

Thanks.
 
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  • #2
f(z) is analytic at infinity if f(1/z) is analytic at zero. Probably the same kind of definition can be given for "bounded at infinity" - but to be sure, you'll need to give us some context. I guess the Banach spaces you're working with are some kind of function spaces.

Anyway, this kind of stuff should be viewed in the context of doing complex analysis on the Riemann sphere (= the complex plane plus a "point at infinity"). This might help you in your search for appropriate reading material.
 
  • #3
morphism said:
f(z) is analytic at infinity if f(1/z) is analytic at zero. Probably the same kind of definition can be given for "bounded at infinity" - but to be sure, you'll need to give us some context. I guess the Banach spaces you're working with are some kind of function spaces.
Thanks, I'm looking at two Banach spaces. An arbitrary infinite-dimensional Banach space X and the Banach space of all bounded linear operators from X to X. Is that enough context? I'm not sure what else to say.
 
  • #4
No, that's not enough context! Where are the analytic functions coming in? Are you trying to define a "functional calculus"? I.e. are you trying to assign meaning to f(T) where f is an analytic function and T is a bounded operator on X?
 
  • #5
Yes that's exactly what I'm trying to do. Sorry should've said that.
 

1. What is "analytic at infinity"?

Analytic at infinity refers to a mathematical concept where a function or equation remains well-defined and behaves predictably as its input values approach infinity.

2. How is a function determined to be analytic at infinity?

A function is considered to be analytic at infinity if it can be expressed as a power series that converges as its input value approaches infinity. This means that the terms in the series become smaller and smaller, and the function approaches a finite value as the input value approaches infinity.

3. What is the significance of a function being analytic at infinity?

If a function is analytic at infinity, it means that it can be used in calculations involving very large or very small values without encountering any unforeseen errors or issues. This makes it a valuable tool in many areas of mathematics and science.

4. Can a function be analytic at some values of infinity but not others?

Yes, a function can be analytic at some values of infinity (such as positive infinity) but not others (such as negative infinity). This means that the function may behave predictably and have a finite value at one end of the number line, but not at the other.

5. How is "analytic at infinity" related to the concept of limits?

The concept of "analytic at infinity" is closely related to the concept of limits, as both involve understanding how a function behaves as its input value approaches a certain value (in this case, infinity). In fact, the limit of a function as its input value approaches infinity is often used to determine if the function is analytic at infinity.

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