Hadamard fractional integral
E259779
The Hadamard fractional integral is a generalization of the classical integral that defines fractional-order integration using logarithmic kernels, particularly suited to functions defined on multiplicative (e.g., positive real) domains.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hadamard fractional integral canonical | 1 |
| Hadamard-type fractional derivative | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2364681 — 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: Hadamard fractional integral Context triple: [Riemann–Liouville integral, comparedWith, Hadamard fractional integral]
-
A.
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.
-
B.
The Fourier Integral and Certain of Its Applications
The Fourier Integral and Certain of Its Applications is a foundational mathematical work by Norbert Wiener that develops and applies Fourier analysis to problems in harmonic analysis and related areas.
-
C.
Hardy–Littlewood maximal function
The Hardy–Littlewood maximal function is a fundamental operator in real analysis and harmonic analysis that controls the local averages of a function and plays a key role in differentiation theorems and singular integral theory.
-
D.
Hardy Z-function
The Hardy Z-function is a real-valued function derived from the Riemann zeta function on the critical line, used extensively in the study of the distribution of its zeros and the Riemann Hypothesis.
-
E.
Selberg integral
The Selberg integral is a fundamental multidimensional generalization of Euler’s beta integral that plays a central role in random matrix theory, combinatorics, and special functions.
- 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: Hadamard fractional integral Target entity description: The Hadamard fractional integral is a generalization of the classical integral that defines fractional-order integration using logarithmic kernels, particularly suited to functions defined on multiplicative (e.g., positive real) domains.
-
A.
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.
-
B.
The Fourier Integral and Certain of Its Applications
The Fourier Integral and Certain of Its Applications is a foundational mathematical work by Norbert Wiener that develops and applies Fourier analysis to problems in harmonic analysis and related areas.
-
C.
Hardy–Littlewood maximal function
The Hardy–Littlewood maximal function is a fundamental operator in real analysis and harmonic analysis that controls the local averages of a function and plays a key role in differentiation theorems and singular integral theory.
-
D.
Hardy Z-function
The Hardy Z-function is a real-valued function derived from the Riemann zeta function on the critical line, used extensively in the study of the distribution of its zeros and the Riemann Hypothesis.
-
E.
Selberg integral
The Selberg integral is a fundamental multidimensional generalization of Euler’s beta integral that plays a central role in random matrix theory, combinatorics, and special functions.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
fractional integral
ⓘ
mathematical concept ⓘ operator ⓘ |
| actsOn |
complex-valued functions
ⓘ
real-valued functions ⓘ |
| appearsIn | theory of Mellin transforms ⓘ |
| connectedTo |
Mellin convolution
ⓘ
linked to:
Mellin transforms
|
| contrastsWith |
Riemann–Liouville integral on additive domains
ⓘ
linked to:
Riemann–Liouville integral
|
| coordinateType | logarithmic scale ⓘ |
| defines | fractional-order integration ⓘ |
| domain |
multiplicative groups
ⓘ
positive real numbers ⓘ |
| field |
fractional calculus
ⓘ
mathematical analysis ⓘ |
| generalizes |
Riemann integral
ⓘ
classical integral ⓘ |
| hasInverse |
Hadamard-type fractional derivative
ⓘ
linked to:
Hadamard fractional integral
|
| hasParameter |
lower limit a > 0
ⓘ
order α > 0 ⓘ |
| hasProperty |
reduces to identity operator when order tends to 0
ⓘ
reduces to repeated classical integral for integer orders ⓘ |
| introducedIn | early 20th century ⓘ |
| invariantUnder | multiplicative scaling of the variable ⓘ |
| isNonlocal | true ⓘ |
| kernelDependsOn | logarithm of the ratio t/x ⓘ |
| namedAfter | Jacques Hadamard ⓘ |
| notation |
H^{α}_{a+} f(x)
ⓘ
I_{a+}^{α,H} f(x) ⓘ |
| orderType |
fractional order
ⓘ
real order ⓘ |
| relatedTo |
Caputo fractional derivative
ⓘ
linked to:
Caputo derivative
Hadamard fractional derivative ⓘ Riemann–Liouville fractional integral ⓘ
linked to:
Riemann–Liouville integral
|
| requiresCondition | integrability of f with logarithmic weight ⓘ |
| satisfies |
linearity
ⓘ
semigroup property in the order parameter ⓘ |
| specialCaseOf | fractional integral with respect to functions ⓘ |
| suitableFor |
functions defined on multiplicative domains
ⓘ
functions defined on positive real axis ⓘ |
| usedFor | modeling memory effects on multiplicative time scales ⓘ |
| usedIn |
differential equations of fractional order
ⓘ
integral equations ⓘ scaling-invariant problems ⓘ |
| usesKernelType | logarithmic kernel ⓘ |
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: Hadamard fractional integral Description of subject: The Hadamard fractional integral is a generalization of the classical integral that defines fractional-order integration using logarithmic kernels, particularly suited to functions defined on multiplicative (e.g., positive real) domains.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Hadamard fractional integral