Riemann–Liouville integral
E47352
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.
All labels observed (6)
How this entity was disambiguated
This entity first appeared as the object of triple T373789 — 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: Riemann–Liouville integral Context triple: [Bernhard Riemann, knownFor, Riemann–Liouville integral]
-
A.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
-
B.
Gaussian integral
The Gaussian integral is a fundamental result in mathematics that evaluates the integral of the exponential of a negative quadratic function over the entire real line, yielding a value proportional to the square root of π and underpinning the normal distribution in probability theory.
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C.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
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D.
Fokker–Planck equation
The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of a stochastic (random) process, such as Brownian motion.
-
E.
Itô calculus
Itô calculus is a branch of stochastic analysis that extends classical calculus to functions of stochastic processes, particularly Brownian motion, enabling rigorous treatment of stochastic differential equations.
- 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: Riemann–Liouville integral Target entity description: 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.
-
A.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
-
B.
Gaussian integral
The Gaussian integral is a fundamental result in mathematics that evaluates the integral of the exponential of a negative quadratic function over the entire real line, yielding a value proportional to the square root of π and underpinning the normal distribution in probability theory.
-
C.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
D.
Fokker–Planck equation
The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of a stochastic (random) process, such as Brownian motion.
-
E.
Itô calculus
Itô calculus is a branch of stochastic analysis that extends classical calculus to functions of stochastic processes, particularly Brownian motion, enabling rigorous treatment of stochastic differential equations.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
concept in fractional calculus
ⓘ
fractional integral operator ⓘ mathematical operator ⓘ |
| actsOn |
locally integrable functions
ⓘ
suitable functions ⓘ |
| appearsIn |
fractional Sobolev spaces
ⓘ
theory of fractional-order systems ⓘ |
| assumes | sufficient regularity of the integrand ⓘ |
| belongsTo | integral transforms with weakly singular kernels ⓘ |
| comparedWith |
Hadamard fractional integral
ⓘ
Weyl fractional integral ⓘ |
| domain | interval [a,b] ⓘ |
| field |
fractional calculus
ⓘ
mathematical analysis ⓘ |
| generalizes |
Cauchy formula for repeated integration
ⓘ
n-fold repeated integral ⓘ |
| hasLimitingCase | identity operator when α → 0^+ ⓘ |
| hasNotation |
I_{-b}^{α}
ⓘ
I_{a+}^{α} ⓘ |
| hasOrder | real order α > 0 ⓘ |
| hasVariant |
Riemann–Liouville integral
self-linksurface differs
ⓘ
surface form:
left-sided Riemann–Liouville integral
Riemann–Liouville integral self-linksurface differs ⓘ
surface form:
right-sided Riemann–Liouville integral
|
| introducedIn | 19th century ⓘ |
| isDefinedBy | convolution with power-law kernel ⓘ |
| isLinear | true ⓘ |
| isToolFor |
defining fractional derivatives
ⓘ
solving fractional integral equations ⓘ |
| kernelType | power-law kernel (x-t)^{α-1} ⓘ |
| namedAfter |
Bernhard Riemann
ⓘ
Joseph Liouville ⓘ |
| parameter |
lower limit a
ⓘ
order α ⓘ |
| property |
depends on entire past history from a to x
ⓘ
nonlocal operator ⓘ |
| reducesTo | n-fold classical integral when α is a positive integer n ⓘ |
| relatedTo |
Caputo derivative
ⓘ
Riemann–Liouville integral self-linksurface differs ⓘ
surface form:
Riemann–Liouville derivative
fractional differential equations ⓘ |
| satisfies | semigroup property in the order α under suitable conditions ⓘ |
| usedFor | defining fractional powers of operators ⓘ |
| usedIn |
anomalous diffusion models
ⓘ
control theory ⓘ modeling memory effects ⓘ signal processing ⓘ viscoelasticity ⓘ |
| usesFunction | Gamma function ⓘ |
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: Riemann–Liouville integral Description of subject: 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.
Referenced by (9)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
left-sided Riemann–Liouville integral
this entity surface form:
right-sided Riemann–Liouville integral
this entity surface form:
Riemann–Liouville derivative
this entity surface form:
Riemann–Liouville derivative
this entity surface form:
Riemann–Liouville fractional integral
this entity surface form:
Riemann–Liouville fractional integral
this entity surface form:
Riemann–Liouville integral on additive domains