Asymptotic Form of an Integral

There is no general answer to this.In summary, the conversation revolved around an integral of the form ##\int^{\infty}_{-\infty}\frac{x(\omega)}{\sigma^{2} + \omega^{2}}d\omega## and whether it is a candidate for asymptotic analysis. The individual asking the question considered using contour methods to evaluate the integral, but was also interested in finding the most general result possible. The two main questions were whether the integral is suitable for asymptotic analysis and if so, where to find the asymptotic form.
  • #1
thrillhouse86
80
0
Hey:

I have an integral of the form:
[tex]
\int^{\infty}_{-\infty}\frac{x(\omega)}{\sigma^{2} + \omega^{2}}d\omega
[/tex]

I'm wondering if this integral is a candidate for asymptotic analysis. My rationale is that as omega increases to either positive infinity or negative infinity, the function being integrated will go to zero. The problem is that every integral I've seen put into an Asymptotic form has had a decaying exponential (which I guess is a much quicker convergence to zero than 1/w^2).

Also - I can guarantee that any f(w) considered is square integrable, and sigma is real

I realize that I can probably just evaluate the integral using contour methods if I have a specific f(w), but I am trying to derive the most general result I can.

I guesss my question is:
1. Is this a candidate for asymptotic analysis
2. If so can someone point me in the direction of obtaining the asymptotic form ?

Thanks
 
Last edited:
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  • #2
Without any knowledge about ##x(\omega)## the integral could be anything.
 

Related to Asymptotic Form of an Integral

What is the asymptotic form of an integral?

The asymptotic form of an integral is the behavior of the integral as the limit of one of its variables approaches a specific value, typically infinity or zero. It is used to study the long-term behavior of a function and is often denoted by the symbol "∼".

What is the significance of the asymptotic form of an integral?

The asymptotic form of an integral is important in many areas of mathematics and science, including calculus, analysis, and statistics. It allows us to understand the behavior of a function as its input approaches a certain value, which can help in making predictions and solving problems.

How is the asymptotic form of an integral calculated?

The asymptotic form of an integral can be calculated using various methods, depending on the specific function and limit being studied. Some common techniques include using L'Hopital's rule, Taylor series expansions, and integration by parts. It is important to consider the properties and limitations of each method in order to accurately determine the asymptotic form.

What are some real-world applications of the asymptotic form of an integral?

The asymptotic form of an integral has a wide range of applications, including in physics, engineering, and economics. For example, it can be used to analyze the long-term behavior of a system, such as the population growth of a species, or the decay of a radioactive substance. It can also be used to approximate solutions to differential equations, and to evaluate the efficiency of algorithms in computer science.

Can the asymptotic form of an integral be used to solve definite integrals?

Yes, the asymptotic form of an integral can be used to approximate definite integrals, particularly when the upper limit of integration is large. This is because the asymptotic form focuses on the behavior of the integral at infinity, which can be easier to evaluate than the integral itself. However, it is important to note that this method may not always provide accurate results and should be used with caution.

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