Cauchy principal value

E239294

mathematical concept method in mathematical analysis

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.

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All labels observed (2)

Label	Occurrences
Cauchy principal value canonical	2
Hadamard finite part	1

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	Cauchy principal value self-linksurface differs ⓘ surface form: Hadamard finite part 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 ⓘ

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Instruction

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# Requirements
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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.

Augustin-Louis Cauchy → knownFor → Cauchy principal value ⓘ

Augustin-Louis → notableFor → Cauchy principal value ⓘ

subject surface form: Augustin-Louis Cauchy

Cauchy principal value → relatedConcept → Cauchy principal value self-linksurface differs ⓘ

this entity surface form: Hadamard finite part