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

Situations with integration over simple poles?

  1. Mar 21, 2014 #1
    This topic is not an application of the ordinary Residue/Cauchy-Riemann theorem, this is a search for any integral occurring within physics (or statistics/math which aren't tailored examples, but that's not the focus for my participation on this forum), which fulfills certain conditions:

    Do you know of any integral;
    ∫f(x)dx over a to b, i.e. a finite interval, which fulfills the following three requirements:

    Suppose that the following conditions are satisfied:
    1 The function f is holomorphic in the extended plane, except for in a finite amount of singularities.
    2. On the interval (a,b) of the real axis f may only have simple poles as singularities.
    3. f has no singularities at {a,b}.

    For representation in latex, see:
    Mathoverflow
    Theorem found in (for those curious):
    Dragoslav S. Mitrinović and Jovan D. Kecić , The Cauchy Method of Residues , 1984 , D. Reidel Publishing Company, theorem 1, chapter 5.4.2, pages 184-185.

    Definition of holomorphic:
    A complex-valued function f(z) is said to be holomorphic on an open set G if it has a derivative at every point of G.

    Definition of extended plane:
    The extended plane is C∪∞.

    Definition of isolated singularity:
    An isolated singularity of f is a point z0 such that fis holomorphic in some punctured disk 0<|z−z0|<R but not holomorphic at z0 itself.

    Definition of simple pole:
    A simple pole is an isolated singularity which can be written as f(z)=1z−z0∗g(z) where g(z) is holomorphic and z0 is the point where the simple pole lies.


    I reiterate: I'm not looking for any help with application, just a situation, physics among others, where this theorem might be applied.
    If there is anything I can do to explain in further detail what I am searching for, please ask.

    Possible results so far:
    Bayesian networks (statistics/probability theory)
     
  2. jcsd
  3. Mar 21, 2014 #2

    maajdl

    User Avatar
    Gold Member

  4. Mar 24, 2014 #3
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted