Euler’s identity for sine product
E697754
infinite product formula
mathematical identity
result in complex analysis
result in special functions
Euler’s identity for sine product is a classical formula expressing the sine function as an infinite product, foundational in the theory of infinite products and special functions.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Euler’s identity for sine product canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7871643 — 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 identity for sine product Context triple: [Jacobi triple product, generalizes, Euler’s identity for sine product]
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A.
Wallis product
The Wallis product is an infinite product formula for π/2, discovered by John Wallis in the 17th century and notable as one of the earliest infinite product representations of π.
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B.
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.
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C.
Euler’s reflection formula
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.
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D.
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.
-
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: Euler’s identity for sine product Target entity description: Euler’s identity for sine product is a classical formula expressing the sine function as an infinite product, foundational in the theory of infinite products and special functions.
-
A.
Wallis product
The Wallis product is an infinite product formula for π/2, discovered by John Wallis in the 17th century and notable as one of the earliest infinite product representations of π.
-
B.
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.
-
C.
Euler’s reflection formula
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.
-
D.
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.
-
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 (46)
| Predicate | Object |
|---|---|
| instanceOf |
infinite product formula
ⓘ
mathematical identity ⓘ result in complex analysis ⓘ result in special functions ⓘ |
| appearsIn |
textbooks on complex analysis
ⓘ
textbooks on special functions ⓘ treatises on trigonometric series ⓘ |
| assumesPropertyOfSine |
oddness of sine
ⓘ
simple zeros at integers ⓘ |
| attributedTo | Leonhard Euler NERFINISHED ⓘ |
| category | Eulerian formula NERFINISHED ⓘ |
| encodesZerosOf | sine function ⓘ |
| expresses | sine as an infinite product over its zeros ⓘ |
| field |
analysis
ⓘ
complex analysis ⓘ special functions ⓘ theory of infinite products ⓘ |
| hasAlternativeForm | \frac{\sin(\pi z)}{\pi z} = \prod_{n=1}^{\infty} \left(1 - \frac{z^{2}}{n^{2}}\right) ⓘ |
| hasConvergenceDomain | all complex z ⓘ |
| hasFormula | \sin(\pi z) = \pi z \prod_{n=1}^{\infty} \left(1 - \frac{z^{2}}{n^{2}}\right) ⓘ |
| hasMathematicalObjectType |
identity involving entire functions
ⓘ
infinite product over integers ⓘ |
| hasZeroStructure | zeros at integer points z = n for n in \mathbb{Z} ⓘ |
| historicalPeriod | 18th century mathematics ⓘ |
| implies |
entirety of the sine function
ⓘ
growth properties of the sine function ⓘ |
| involvesConstant | \pi ⓘ |
| involvesFunction |
\sin(\pi z)
ⓘ
sine function ⓘ |
| involvesOperation |
infinite product
ⓘ
limit ⓘ multiplication ⓘ |
| involvesVariable | complex variable z ⓘ |
| motivated | development of infinite product representations of analytic functions ⓘ |
| productIndex | n from 1 to infinity ⓘ |
| relatedTo |
Euler’s product for the gamma function
NERFINISHED
ⓘ
Weierstrass product for the sine function NERFINISHED ⓘ reflection formula for the gamma function ⓘ |
| specialCase | \sin x = x \prod_{n=1}^{\infty} \left(1 - \frac{x^{2}}{n^{2}\pi^{2}}\right) ⓘ |
| typeOf | canonical product representation ⓘ |
| usedIn |
Weierstrass factorization theory
NERFINISHED
ⓘ
derivation of product expansions for trigonometric functions ⓘ derivation of product formula for \frac{\sin(\pi z)}{\pi z} ⓘ theory of entire functions ⓘ |
| usedToDerive |
identities involving \zeta(2n)
ⓘ
relations between trigonometric and zeta values ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Euler’s identity for sine product Description of subject: Euler’s identity for sine product is a classical formula expressing the sine function as an infinite product, foundational in the theory of infinite products and special functions.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.