Gauss multiplication formula
E596521
The Gauss multiplication formula is a classical identity in complex analysis that expresses the gamma function of a multiple of a variable as a product of gamma functions evaluated at shifted fractions of that variable.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Euler’s product for the gamma function | 1 |
| Gauss multiplication formula canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6482559 — 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: Gauss multiplication formula Context triple: [Gamma function, hasMultiplicationFormula, Gauss multiplication formula]
-
A.
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.
-
B.
Pochhammer symbol
The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
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C.
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.
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D.
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.
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E.
Jacobi triple product
The Jacobi triple product is a fundamental identity in number theory and complex analysis that expresses an infinite product as an infinite sum, playing a key role in the theory of theta functions and q-series.
- 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: Gauss multiplication formula Target entity description: The Gauss multiplication formula is a classical identity in complex analysis that expresses the gamma function of a multiple of a variable as a product of gamma functions evaluated at shifted fractions of that variable.
-
A.
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.
-
B.
Pochhammer symbol
The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
-
C.
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.
-
D.
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.
-
E.
Jacobi triple product
The Jacobi triple product is a fundamental identity in number theory and complex analysis that expresses an infinite product as an infinite sum, playing a key role in the theory of theta functions and q-series.
- F. None of above. chosen
Statements (39)
| Predicate | Object |
|---|---|
| instanceOf |
formula involving the gamma function
ⓘ
identity in complex analysis ⓘ mathematical formula ⓘ |
| appearsIn |
textbooks on complex analysis
ⓘ
textbooks on special functions ⓘ treatises on the gamma function ⓘ |
| category |
gamma function identities
ⓘ
multiplication theorems in analysis ⓘ |
| domainVariable | complex variable z ⓘ |
| expresses | gamma function of a multiple of a variable as a product of shifted gamma functions ⓘ |
| field |
complex analysis
ⓘ
mathematical analysis ⓘ special functions ⓘ |
| givesExpressionFor | Γ(nz) ⓘ |
| hasProperty |
extends to meromorphic identity on ℂ
ⓘ
provides finite product representation for Γ(nz) in terms of Γ at shifted arguments ⓘ |
| involvesFunction | gamma function ⓘ |
| isGeneralizationOf |
duplication formula for the gamma function
ⓘ
triplication formula for the gamma function ⓘ |
| namedAfter | Carl Friedrich Gauss ⓘ |
| parameter | positive integer n ⓘ |
| relatedTo |
Euler reflection formula
ⓘ
linked to:
Euler’s reflection formula
Weierstrass product for the gamma function ⓘ
linked to:
Weierstrass factorization theorem
|
| relates |
Γ(nz)
ⓘ
Γ(z) ⓘ Γ(z+(n-1)/n) ⓘ Γ(z+1/n) ⓘ Γ(z+2/n) ⓘ |
| requiresCondition | n is a positive integer for the standard form ⓘ |
| usedIn |
analytic number theory
ⓘ
asymptotic analysis of special functions ⓘ derivations involving the beta function ⓘ evaluation of certain infinite products ⓘ functional equations for special functions ⓘ |
| usesConstant | (2π)^{(1-n)/2} ⓘ |
| usesFactor | n^{nz-1/2} ⓘ |
| usesProduct | ∏_{k=0}^{n-1} Γ(z + k/n) ⓘ |
| validFor |
n ∈ ℕ, n ≥ 1
ⓘ
z ∈ ℂ with Γ defined and finite ⓘ |
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: Gauss multiplication formula Description of subject: The Gauss multiplication formula is a classical identity in complex analysis that expresses the gamma function of a multiple of a variable as a product of gamma functions evaluated at shifted fractions of that variable.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Gauss multiplication formula