Gamma function
E146428
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Gamma function canonical | 16 |
| Gamma function identities | 1 |
| gamma function | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1286007 — 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: Gamma function Context triple: [generalized binomial theorem, relatedConcept, Gamma function]
-
A.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
B.
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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C.
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.
-
D.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
E.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
- 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: Gamma function Target entity description: 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.
-
A.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
B.
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.
-
C.
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.
-
D.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
E.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
extension of factorial
ⓘ
meromorphic function ⓘ special function ⓘ |
| agreesWithFactorialOn | Γ(n+1)=n! for n∈ℕ ⓘ |
| alsoDevelopedBy | Adrien-Marie Legendre ⓘ |
| appearsIn |
Student’s t-distribution
ⓘ
beta distribution ⓘ chi-square distribution ⓘ gamma distribution ⓘ |
| codomain | complex numbers ⓘ |
| containsConstant | Euler–Mascheroni constant γ ⓘ |
| definedOn | complex numbers except non-positive integers ⓘ |
| domain | complex plane minus non-positive integers ⓘ |
| generalizes | factorial function to non-integers ⓘ |
| growthOrder | order 1 in complex plane ⓘ |
| hasDuplicationFormula | Γ(z)Γ(z+1/2)=2^{1-2z}√π Γ(2z) ⓘ |
| hasIntegralRepresentation | Γ(z)=∫₀^∞ t^{z-1}e^{-t}dt for Re(z)>0 ⓘ |
| hasLogarithmicDerivative | digamma function ⓘ |
| hasMultiplicationFormula | Gauss multiplication formula ⓘ |
| hasReflectionFormula | Γ(z)Γ(1−z)=π/sin(πz) ⓘ |
| hasSimplePolesAt | z=0,-1,-2,… ⓘ |
| hasWeierstrassProduct | 1/Γ(z)=ze^{γz}∏_{n=1}^∞(1+z/n)e^{-z/n} ⓘ |
| introducedBy | Leonhard Euler ⓘ |
| isEvenOrOdd | neither even nor odd ⓘ |
| isHolomorphicOn | ℂ minus non-positive integers ⓘ |
| isLogarithmicallyConvexOn | (0,∞) ⓘ |
| isLogConvexOn | (0,∞) ⓘ |
| nonZeroOn | right half-plane Re(z)>0 ⓘ |
| normalizationConstantFor |
Dirichlet distribution density
ⓘ
linked to:
Dirichlet distribution
beta distribution density ⓘ gamma distribution density ⓘ |
| relatedFunction |
Euler’s reflection formula
ⓘ
beta function ⓘ incomplete gamma function ⓘ polygamma function ⓘ |
| satisfiesFunctionalEquation | Γ(z+1)=zΓ(z) ⓘ |
| satisfiesRecurrence | Γ(z+1)=zΓ(z) ⓘ |
| satisfiesStirlingApproximation | Γ(z)~√(2π) z^{z-1/2} e^{-z} as |z|→∞ in sector ⓘ |
| symbol | Γ(z) ⓘ |
| usedIn |
asymptotic analysis
ⓘ
complex analysis ⓘ mathematical physics ⓘ number theory ⓘ probability theory ⓘ representation of distributions ⓘ statistics ⓘ |
| valueAt |
Γ(1)=1
ⓘ
Γ(1/2)=√π ⓘ Γ(n)=(n-1)! for n∈ℕ ⓘ |
| yearIntroducedApprox | 18th century ⓘ |
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: Gamma function Description of subject: 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.
Referenced by (18)
Full triples — surface form annotated when it differs from this entity's canonical label.