Euler’s reflection formula
E596522
Euler’s reflection formula is a fundamental identity in complex analysis that relates the values of the Gamma function at z and 1−z through the sine function, revealing a deep symmetry of the Gamma function.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Gamma function reflection formula | 2 |
| Euler reflection formula | 1 |
| Euler reflection formula for Gamma function | 1 |
| Euler’s reflection formula canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6482564 — 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: Euler’s reflection formula Context triple: [Gamma function, relatedFunction, Euler’s reflection formula]
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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.
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.
-
C.
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.
-
D.
Euler’s formula for complex exponentials
Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
-
E.
Euler product formula for the Riemann zeta function
The Euler product formula for the Riemann zeta function is a fundamental identity in analytic number theory that expresses the zeta function as an infinite product over all prime numbers, revealing a deep connection between primes and the distribution of integers.
- 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: Euler’s reflection formula Target entity description: Euler’s reflection formula is a fundamental identity in complex analysis that relates the values of the Gamma function at z and 1−z through the sine function, revealing a deep symmetry of the Gamma function.
-
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.
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.
-
C.
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.
-
D.
Euler’s formula for complex exponentials
Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
-
E.
Euler product formula for the Riemann zeta function
The Euler product formula for the Riemann zeta function is a fundamental identity in analytic number theory that expresses the zeta function as an infinite product over all prime numbers, revealing a deep connection between primes and the distribution of integers.
- F. None of above. chosen
Statements (31)
| Predicate | Object |
|---|---|
| instanceOf | mathematical identity ⓘ |
| appearsIn |
analytic number theory
ⓘ
complex analysis textbooks ⓘ theory of special functions ⓘ |
| classification | reflection formula for special functions ⓘ |
| connects | Gamma function and trigonometric functions ⓘ |
| domainCondition |
z not an integer
ⓘ
z ∉ ℤ ⓘ |
| field |
complex analysis
ⓘ
special functions ⓘ |
| hasAlternativeForm |
Γ(z) Γ(1 - z) = π csc(πz)
ⓘ
Γ(z) Γ(1 - z) sin(πz) = π ⓘ |
| hasExpression | Γ(z) Γ(1 - z) = π / sin(πz) ⓘ |
| implies |
zeros of sin(πz) correspond to poles of Γ(z) Γ(1 - z)
ⓘ
Γ(z) Γ(1 - z) is meromorphic with simple poles at integers ⓘ |
| involvesFunction |
Gamma function
ⓘ
sine function ⓘ |
| namedAfter | Leonhard Euler ⓘ |
| relatedTo |
Euler’s formula for the sine function via infinite product
ⓘ
linked to:
Euler’s identity for sine product
Gamma function functional equation Γ(z+1) = z Γ(z) ⓘ Riemann zeta function functional equations ⓘ |
| relates |
sin(πz)
ⓘ
Γ(1 - z) ⓘ Γ(z) ⓘ |
| revealsPropertyOf | symmetry of the Gamma function ⓘ |
| typeOfSymmetry | reflection across the line Re(z) = 1/2 ⓘ |
| usedFor |
analytic continuation of the Gamma function
ⓘ
deriving identities for trigonometric functions ⓘ relating Γ(z) and Γ(1 - z) ⓘ studying functional equations of special functions ⓘ |
| validOn | complex plane minus the 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: Euler’s reflection formula Description of subject: Euler’s reflection formula is a fundamental identity in complex analysis that relates the values of the Gamma function at z and 1−z through the sine function, revealing a deep symmetry of the Gamma function.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Euler’s reflection formula
linked to: Euler’s reflection formula
linked to: Euler’s reflection formula
linked to: Euler’s reflection formula