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Elendur
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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)
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)
