Find Cauchy Principal Value of Improper Integral

In summary, determining the convergence of an improper integral involving a singularity requires considering three separate regions and evaluating the limit as the bound approaches the singularity. The singularity is non-integrable if this limit does not exist, and it can often be determined by observing the asymptotic behavior of the integrand.
  • #1
Mr Davis 97
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Say I am given the integral ##\displaystyle \int_0^{\infty} \frac{x \sin (ax)}{x^2 - b^2} dx##. How can I determine whether this improper integral converges in the normal sense, or whether I should just look for the Cauchy Principal Value?
 
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  • #2
Mr Davis 97 said:
How can I determine whether this improper integral converges in the normal sense
There are three regions for x where there could be a problem. Consider each separately.
 
  • #3
Singularity (not integrable) at |x-b|=0. Need Principal value.
 
  • #4
mathman said:
Singularity (not integrable) at |x-b|=0. Need Principal value.
Why does there being a singularity mean that it is not integrable? Couldn't we try to evaluate it as any other improper integral, using limits on the integral as x approaches b and seeing if it converges?
 
  • #5
Mr Davis 97 said:
Why does there being a singularity mean that it is not integrable?
I believe mathman is saying it is a non-integrable singularity. It is equivalent to integrating 1/x through 0.
 
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  • #6
haruspex said:
I believe mathman is saying it is a non-integrable singularity. It is equivalent to integrating 1/x through 0.
Would it only be an integrable singularity of it were a removable or jump discontinuity?
 
  • #7
Mr Davis 97 said:
Would it only be an integrable singularity of it were a removable or jump discontinuity?
No, you can integrate x-0.5 through 0. The limit as a tends to zero of ∫abx-0.5.dx exists.
 
  • #8
haruspex said:
No, you can integrate x-0.5 through 0. The limit as a tends to zero of ∫abx-0.5.dx exists.
So when exactly is a singularity non-integrable?
 
  • #9
Mr Davis 97 said:
So when exactly is a singularity non-integrable?
When the limit, as a bound approaches the singularity, does not exist. That's when, in order to integrate through the singularity, you have to play questionable games letting the negative infinities on one side cancel the positive infinities on the other.
 
  • #10
haruspex said:
When the limit, as a bound approaches the singularity, does not exist. That's when, in order to integrate through the singularity, you have to play questionable games letting the negative infinities on one side cancel the positive infinities on the other.
Can I tell just from looking at the integral whether the limit, as a bound approaches a singularity, does or does not exist? Or do I explicit need evaluate the integral to the point where I can show that the limit does not exist?
 
  • #11
Mr Davis 97 said:
Can I tell just from looking at the integral whether the limit, as a bound approaches a singularity, does or does not exist? Or do I explicit need evaluate the integral to the point where I can show that the limit does not exist?
You can usually spot the asymptotic behaviour, rather than integrating exactly. For (x2-b2)-1, you can factor out the x+b and observe that it will be roughly 2b in the neighbourhood of x=b. Therefore it converges if and only the integral of (x-b)-1 does.
 
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1. What is an improper integral?

An improper integral is an integral that does not have a finite value due to one or both of its limits being infinite, or the function being undefined at a certain point within the limits.

2. What is the Cauchy principal value of an improper integral?

The Cauchy principal value of an improper integral is the value obtained by taking the average of the two one-sided limits as the limits of integration approach the undefined point.

3. How do you find the Cauchy principal value of an improper integral?

To find the Cauchy principal value, you must first identify the point of discontinuity or infinity within the limits of integration. Then, you must evaluate the integral by taking the average of the two one-sided limits as the limits approach the undefined point.

4. Why is it important to find the Cauchy principal value of an improper integral?

Finding the Cauchy principal value allows us to evaluate integrals that would otherwise be undefined or infinite. It provides a way to assign a meaningful value to these types of integrals and can be helpful in solving real-world problems.

5. When is it necessary to use the Cauchy principal value?

The Cauchy principal value is necessary when evaluating integrals that have a point of discontinuity or infinity within their limits. It is also used when the function being integrated is undefined at a certain point within the limits.

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