Cauchy principal value
E239294
The Cauchy principal value is a method in mathematical analysis for assigning finite values to certain improper or divergent integrals and series by symmetrically balancing their singularities.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Cauchy principal value canonical | 2 |
| Hadamard finite part | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2171655 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Cauchy principal value Context triple: [Augustin-Louis Cauchy, knownFor, Cauchy principal value]
-
A.
Fresnel integrals
Fresnel integrals are special functions in mathematics that describe the complex oscillatory behavior of wave diffraction and interference, particularly in optics.
-
B.
Riemann–Liouville integral
The Riemann–Liouville integral is a fundamental operator in fractional calculus that generalizes the concept of an n-fold repeated integral to non-integer (fractional) orders.
-
C.
Cauchy-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
-
D.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
E.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Cauchy principal value Target entity description: The Cauchy principal value is a method in mathematical analysis for assigning finite values to certain improper or divergent integrals and series by symmetrically balancing their singularities.
-
A.
Fresnel integrals
Fresnel integrals are special functions in mathematics that describe the complex oscillatory behavior of wave diffraction and interference, particularly in optics.
-
B.
Riemann–Liouville integral
The Riemann–Liouville integral is a fundamental operator in fractional calculus that generalizes the concept of an n-fold repeated integral to non-integer (fractional) orders.
-
C.
Cauchy-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
-
D.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
E.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
method in mathematical analysis ⓘ |
| abbreviation |
PV
ⓘ
p.v. ⓘ |
| appliesTo |
divergent integrals
ⓘ
divergent series ⓘ improper integrals ⓘ singular integrals ⓘ |
| characteristic |
balances singularities symmetrically
ⓘ
can assign finite value to divergent expressions ⓘ depends on the way limits are taken ⓘ is not absolutely convergent in general ⓘ uses symmetric limiting process ⓘ |
| context |
appears in solving singular Cauchy-type integral equations
ⓘ
often denoted by a dash through the integral sign ⓘ used in defining boundary values of analytic functions ⓘ used to interpret integrals with poles on the path of integration ⓘ |
| definitionExample |
PV ∫_{-a}^{a} f(x) dx = lim_{ε→0+} (∫_{-a}^{-ε} f(x) dx + ∫_{ε}^{a} f(x) dx)
ⓘ
PV ∫_{-∞}^{∞} f(x) dx = lim_{R→∞} ∫_{-R}^{R} f(x) dx ⓘ PV ∫_{a}^{b} f(x) dx with singularity at c is lim_{ε→0+} (∫_{a}^{c-ε} f(x) dx + ∫_{c+ε}^{b} f(x) dx) ⓘ |
| example | PV ∫_{-1}^{1} (1/x) dx = 0 ⓘ |
| field |
complex analysis
ⓘ
distribution theory ⓘ mathematical analysis ⓘ real analysis ⓘ |
| namedAfter | Augustin-Louis Cauchy ⓘ |
| property |
coincides with usual integral when the integral converges absolutely
ⓘ
is invariant under odd symmetry for integrals over symmetric intervals ⓘ is linear where defined ⓘ may exist when the usual improper integral does not exist ⓘ |
| relatedConcept |
Hadamard finite part
ⓘ
linked to:
Cauchy principal value
Hilbert transform ⓘ conditional convergence ⓘ distribution (generalized function) ⓘ improper integral ⓘ principal value distribution of 1/x ⓘ |
| usedIn |
Fourier analysis
ⓘ
Hilbert transform ⓘ Kramers–Kronig relations ⓘ complex contour integration ⓘ dispersion relations ⓘ distribution theory of tempered distributions ⓘ principal value integrals in physics ⓘ quantum field theory regularization techniques ⓘ regularization of divergent integrals ⓘ singular integral equations ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Cauchy principal value Description of subject: The Cauchy principal value is a method in mathematical analysis for assigning finite values to certain improper or divergent integrals and series by symmetrically balancing their singularities.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.