Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integrable singularity

  1. Mar 5, 2008 #1

    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)
  2. jcsd
  3. Mar 5, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    An integrable singularity is such that
    1. When an arbitrary neighbourhood of the singularity is excluded the integral is well.defined.
    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:

    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?