Barnes G-function
E865106
The Barnes G-function is a special function that generalizes the superfactorial and is closely related to the gamma and multiple gamma functions in complex analysis and number theory.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Barnes G-function canonical | 1 |
| Barnes functions | 1 |
| K-function of Kinkelin | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10462102 — 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: Barnes G-function Context triple: [Selberg integral, relatedTo, Barnes G-function]
-
A.
Gamma function
The Gamma function is a fundamental extension of the factorial function to complex and real non-integer arguments, widely used in analysis, probability, and mathematical physics.
-
B.
Gauss hypergeometric function
The Gauss hypergeometric function is a special function defined by a power series that generalizes many elementary and higher transcendental functions and plays a central role in mathematical analysis, differential equations, and mathematical physics.
-
C.
Beta function
The Beta function is a special function in mathematics, closely related to the Gamma function, that arises in calculus, probability theory, and complex analysis, particularly in evaluating integrals and expressing various identities.
-
D.
Mittag-Leffler function
The Mittag-Leffler function is a complex function that generalizes the exponential function and plays a central role in fractional calculus and the theory of differential and integral equations.
-
E.
Bessel functions
Bessel functions are special mathematical functions that commonly arise as solutions to differential equations with cylindrical symmetry, widely used in physics and engineering.
- 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: Barnes G-function Target entity description: The Barnes G-function is a special function that generalizes the superfactorial and is closely related to the gamma and multiple gamma functions in complex analysis and number theory.
-
A.
Gamma function
The Gamma function is a fundamental extension of the factorial function to complex and real non-integer arguments, widely used in analysis, probability, and mathematical physics.
-
B.
Gauss hypergeometric function
The Gauss hypergeometric function is a special function defined by a power series that generalizes many elementary and higher transcendental functions and plays a central role in mathematical analysis, differential equations, and mathematical physics.
-
C.
Beta function
The Beta function is a special function in mathematics, closely related to the Gamma function, that arises in calculus, probability theory, and complex analysis, particularly in evaluating integrals and expressing various identities.
-
D.
Mittag-Leffler function
The Mittag-Leffler function is a complex function that generalizes the exponential function and plays a central role in fractional calculus and the theory of differential and integral equations.
-
E.
Bessel functions
Bessel functions are special mathematical functions that commonly arise as solutions to differential equations with cylindrical symmetry, widely used in physics and engineering.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
complex-analytic function
ⓘ
entire function ⓘ special function ⓘ |
| appearsIn |
evaluation of certain infinite products
ⓘ
multiple zeta and multiple gamma theory ⓘ |
| category |
Barnes functions
ⓘ
linked to:
Barnes G-function
|
| domain | complex plane ⓘ |
| field |
complex analysis
ⓘ
number theory ⓘ |
| generalizes |
factorial-related products
ⓘ
superfactorial ⓘ |
| growthType | order 2 entire function ⓘ |
| hasAsymptoticExpansion | log G(z+1) expressed using Bernoulli numbers ⓘ |
| hasLogDerivative | log G(z+1)= (z/2)log(2π) - (3z^2/4) + O(z log z) as z→∞ (up to constants) ⓘ |
| hasProperty | log-convex on positive real axis ⓘ |
| hasWeierstrassProduct | infinite product representation involving Γ(z) ⓘ |
| hasZeroAt |
z=-1
ⓘ
z=-2 ⓘ z=-3 ⓘ z=0 ⓘ |
| introducedBy | Ernest William Barnes ⓘ |
| introducedIn | early 20th century ⓘ |
| isEntireIn | z ⓘ |
| isSolutionOf | functional equation of second order in shifts ⓘ |
| namedAfter | Ernest William Barnes ⓘ |
| normalization | G(1)=1 ⓘ |
| notation | G(z) ⓘ |
| relatedTo |
Barnes multiple zeta function
ⓘ
K-function of Kinkelin ⓘ
linked to:
Barnes G-function
Riemann zeta function ⓘ gamma function ⓘ multiple gamma function ⓘ superfactorial function n$!$ (product of factorials) ⓘ |
| satisfies | G(z)≠0 for non-integer complex z ⓘ |
| satisfiesFunctionalEquation | G(z+1)=Γ(z)G(z) ⓘ |
| satisfiesRecurrence | G(n+1)=Γ(n)G(n) for positive integers n ⓘ |
| specialValue | G(n+1)=∏_{k=1}^{n-1}k! for positive integers n ⓘ |
| usedFor |
evaluation of partition functions in physics
ⓘ
regularization of infinite products ⓘ |
| usedIn |
asymptotic analysis
ⓘ
determinants of Laplacians ⓘ quantum field theory ⓘ random matrix theory ⓘ spectral theory ⓘ |
| valueAt |
G(1)=1
ⓘ
G(2)=1 ⓘ G(3)=1 ⓘ |
| zeroSet | non-positive integers ⓘ |
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: Barnes G-function Description of subject: The Barnes G-function is a special function that generalizes the superfactorial and is closely related to the gamma and multiple gamma functions in complex analysis and number theory.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Barnes G-function
linked to: Barnes G-function