Register to reply

Situations with integration over simple poles?

by Elendur
Tags: integration, poles, simple, situations
Share this thread:
Mar21-14, 05:49 AM
P: 2
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:
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)
Phys.Org News Partner Physics news on
Vibrational motion of a single molecule measured in real time
Researchers demonstrate ultra low-field nuclear magnetic resonance using Earth's magnetic field
Bubbling down: Discovery suggests surprising uses for common bubbles
Mar21-14, 07:22 AM
PF Gold
P: 370
See the theory of Landau damping.
For example:

(page 9, "The Landau Contour")
Mar24-14, 07:25 AM
P: 2
Quote Quote by maajdl View Post
See the theory of Landau damping.
For example:

(page 9, "The Landau Contour")
Awesome, thank you very much! I'll look into this at the first opportunity.

Register to reply

Related Discussions
Complex Integration Function with multiple poles at origin Calculus & Beyond Homework 6
Complex Integration - Poles on the Imaginary axis Calculus & Beyond Homework 3
Complex Integrals - Poles of Integration Outside the Curve Calculus & Beyond Homework 2
Numerical Integration of Functions with Poles Calculus 1