# Situations with integration over simple poles?

• Elendur
In summary, this conversation discusses the search for any integral that fulfills certain conditions within physics or other fields such as statistics or mathematics. These conditions include the function being holomorphic except for a finite amount of singularities, having only simple poles as singularities on a finite interval, and having no singularities at the endpoints of the interval. The Cauchy-Riemann theorem is mentioned as a possible application in the extended plane, and examples are given in the theory of Landau damping in plasma physics.
Elendur
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:
http://mathoverflow.net/questions/160833/integration-over-a-finite-interval-containing-simple-poles-any-applications
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)

## 1. What is integration over simple poles?

Integration over simple poles is a method used in calculus to evaluate integrals that contain singularities (poles) that are simple, meaning they have a degree of one. This method involves finding the residue of the function at the pole and then using it to evaluate the integral.

## 2. How do you identify simple poles in a function?

Simple poles can be identified by looking at the denominator of the function. If it can be factored into linear terms, then each term will correspond to a simple pole. Additionally, if the denominator has a term raised to a power greater than one, it is not a simple pole.

## 3. Can integration over simple poles be used for all types of functions?

No, integration over simple poles can only be used for functions that have simple poles as singularities. Functions with other types of singularities, such as essential or branch point singularities, require different methods of integration.

## 4. What is the significance of finding the residue in integration over simple poles?

The residue is the coefficient of the term with the simple pole in the partial fraction decomposition of the function. It is used in integration over simple poles to evaluate the integral and can also be used to find the coefficients of the other terms in the decomposition.

## 5. Are there any limitations or challenges to using integration over simple poles?

One limitation is that the function must have simple poles as singularities for this method to work. Additionally, finding the residue can be time-consuming and complex, especially for functions with multiple poles. It also requires a good understanding of partial fraction decomposition and complex analysis.

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