What is an integrable singularity and how is it determined?

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In summary, an integrable singularity refers to a singularity where the integral is well-defined when an arbitrary neighborhood of the singularity is excluded and the limit of the integral approaches a definite value as the size of the neighborhood shrinks to 0. An example of an integrable singularity is the integral of \log{x} from \epsilon to 1, where the singularity is given a unique value of -1 as \epsilon approaches 0.
  • #1
daudaudaudau
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Hello.

What is an integrable singularity? Is it a certain order of the singularity?

[tex]\int_0^1\frac{1}{x}=\infty[/tex] (not integrable)
[tex]\int_0^1\log{x}=-1[/tex] (integrable)
 
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  • #2
An integrable singularity is such that
1. When an arbitrary neighbourhood of the singularity is excluded the integral is well.defined.
and
2. When the size of that neighbourhood is shrunk to 0, we get a definite limit.

For example, let's tackle your second one:

NOw, let e be a number greater than 0, and consider:
[tex]\int_{\epsilon}^{1}\log(x)=-1-\epsilon\log(\epsilon)+\epsilon[/tex]

Since the two last terms go to zero as e goes to 0, we have that the improper integral can be given the unique value -1. The singularity was integrable.
 

Related to What is an integrable singularity and how is it determined?

1. What is an integrable singularity?

An integrable singularity is a mathematical term used to describe a point or set of points where a function is not well-defined or is discontinuous. In other words, the function cannot be integrated at this point.

2. How is an integrable singularity different from a regular singularity?

An integrable singularity is different from a regular singularity in that it can be resolved by changing variables or using a different method of integration, while a regular singularity cannot be resolved and the function is undefined at that point.

3. Can an integrable singularity be removed?

Yes, an integrable singularity can be removed by using a change of variables or a different method of integration. However, this may result in a different solution or representation of the original function.

4. Are integrable singularities common in scientific research?

Yes, integrable singularities are common in scientific research, especially in fields such as physics, engineering, and mathematics. They often arise in the study of complex and nonlinear systems.

5. How can integrable singularities be handled in numerical computations?

Integrable singularities can be handled in numerical computations by using specialized algorithms and techniques that can accurately handle and resolve these points. It is important to identify and properly handle integrable singularities to obtain accurate and meaningful results in numerical computations.

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