Riemann–Liouville integral

E47352

concept in fractional calculus fractional integral operator mathematical operator

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.

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All labels observed (6)

Label	Occurrences
Riemann–Liouville integral canonical	3
Riemann–Liouville derivative	2
Riemann–Liouville fractional integral	2
Riemann–Liouville integral on additive domains	1
left-sided Riemann–Liouville integral	1
right-sided Riemann–Liouville integral	1

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.
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
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	left-sided Riemann–Liouville integral ⓘ linked to: Riemann–Liouville integral right-sided Riemann–Liouville integral ⓘ linked to: 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 derivative ⓘ linked to: Riemann–Liouville integral 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