Defining Analyticity at Infinity: How Do You Define and Calculate It?

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In summary, analyticity at infinity is the behavior of a mathematical function as its input approaches infinity. It is important in understanding the long-term behavior of functions, especially in fields like physics and engineering. To determine if a function is analytic at infinity, its behavior as the input approaches infinity must be analyzed. This is different from analyticity at a finite point, which refers to the behavior of a function at a specific input value. It is possible for a function to be analytic at infinity but not at a finite point, meaning that its behavior is well-defined at infinity but not at a finite input value.
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Palindrom
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How does one define, given a complex function, the following:

  • The function is analytic at infinity.
  • The derivative of the function at infinity.

It turns out that it's supposed to be quite common to define these terms, however I have never been shown either of them. I have a few guesses, but along with some lecture notes I could put my hands on, it's all become a big mess for me.
 
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  • #2
Presumably one replaces z with w=1/z and works w=0.
 
  • #3
O.K., but then if g(z)=f(1/z) do I take f'(\infty)=g'(0)? Just like this?
 

FAQ: Defining Analyticity at Infinity: How Do You Define and Calculate It?

What is analyticity at infinity?

Analyticity at infinity refers to the behavior of a mathematical function as its input approaches infinity. In other words, it describes how the function behaves as its input values become extremely large.

Why is analyticity at infinity important?

Analyticity at infinity is important because it allows us to understand the long-term behavior of a function. This is especially useful in fields such as physics and engineering, where functions may represent physical phenomena that occur over large distances or time scales.

How do you determine if a function is analytic at infinity?

To determine if a function is analytic at infinity, we must analyze its behavior as the input approaches infinity. If the function can be approximated by a polynomial with increasing powers of the input, then it is considered to be analytic at infinity.

What is the difference between analyticity at infinity and analyticity at a finite point?

Analyticity at infinity and analyticity at a finite point refer to different types of behaviors of a function. Analyticity at infinity describes the behavior of a function as its input approaches infinity, while analyticity at a finite point describes the behavior of a function at a specific, finite input value.

Can a function be analytic at infinity but not at a finite point?

Yes, a function can be analytic at infinity but not at a finite point. This means that the function has a well-defined behavior as its input approaches infinity, but may have undefined or discontinuous behavior at a finite input value.

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