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:(adsbygoogle = window.adsbygoogle || []).push({});

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 ofholomorphic:

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 ofextended plane:

The extended plane is C∪∞.

Definition ofisolated 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 ofsimple 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)

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# Situations with integration over simple poles?

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