What is an integrable singularity and how is it determined?

[daudaudaudau]

Hello.

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

$$\int_0^1\frac{1}{x}=\infty$$ (not integrable)
$$\int_0^1\log{x}=-1$$ (integrable)

 

[arildno]

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:
$$\int_{\epsilon}^{1}\log(x)=-1-\epsilon\log(\epsilon)+\epsilon$$

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.