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Situations with integration over simple poles? 
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#1
Mar2114, 05:49 AM

P: 2

This topic is not an application of the ordinary Residue/CauchyRiemann 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 184185. Definition of holomorphic: A complexvalued 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
Mar2114, 07:22 AM

PF Gold
P: 354

See the theory of Landau damping.
For example: http://theory.physics.helsinki.fi/~p...heory_2012.pdf or http://www.pma.caltech.edu/Courses/p...4/0421.1.K.pdf (page 9, "The Landau Contour") 


#3
Mar2414, 07:25 AM

P: 2




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