Hadamard fractional integral

E259779

fractional integral mathematical concept operator

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.

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All labels observed (2)

Label	Occurrences
Hadamard fractional integral canonical	1
Hadamard-type fractional derivative	1

Statements (44)

Predicate	Object
instanceOf	fractional integral ⓘ mathematical concept ⓘ operator ⓘ
actsOn	complex-valued functions ⓘ real-valued functions ⓘ
appearsIn	theory of Mellin transforms ⓘ
connectedTo	Mellin transforms ⓘ surface form: Mellin convolution
contrastsWith	Riemann–Liouville integral ⓘ surface form: Riemann–Liouville integral on additive domains
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 fractional integral self-linksurface differs ⓘ surface form: Hadamard-type fractional derivative
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 derivative ⓘ surface form: Caputo fractional derivative Hadamard fractional derivative ⓘ Riemann–Liouville integral ⓘ surface form: Riemann–Liouville fractional 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 ⓘ

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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.

Riemann–Liouville integral → comparedWith → Hadamard fractional integral ⓘ

Hadamard fractional integral → hasInverse → Hadamard fractional integral self-linksurface differs ⓘ

this entity surface form: Hadamard-type fractional derivative