Wallis product
E587242
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Wallis product canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6355610 — 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: Wallis product Context triple: [John Wallis, notableConcept, Wallis product]
-
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.
Sylvester sequence
The Sylvester sequence is an integer sequence defined recursively where each term is one more than the product of all previous terms, yielding rapidly growing, pairwise coprime numbers closely related to Egyptian fraction representations.
-
C.
Khinchin's constant
Khinchin's constant is a mathematical constant that arises in metric number theory, describing the almost-sure geometric mean of the partial quotients in the continued fraction expansions of real numbers.
-
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: Wallis product Target entity description: 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.
-
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.
Sylvester sequence
The Sylvester sequence is an integer sequence defined recursively where each term is one more than the product of all previous terms, yielding rapidly growing, pairwise coprime numbers closely related to Egyptian fraction representations.
-
C.
Khinchin's constant
Khinchin's constant is a mathematical constant that arises in metric number theory, describing the almost-sure geometric mean of the partial quotients in the continued fraction expansions of real numbers.
-
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 (44)
| Predicate | Object |
|---|---|
| instanceOf |
infinite product
ⓘ
mathematical formula ⓘ representation of pi ⓘ |
| alternativeExpression |
π/2 = (2·2)/(1·3) · (4·4)/(3·5) · (6·6)/(5·7) · …
ⓘ
π/2 = ∏_{n=1}^{∞} (2n·2n)/( (2n-1)(2n+1) ) ⓘ |
| appearsIn |
textbooks on calculus
ⓘ
textbooks on mathematical history ⓘ textbooks on real analysis ⓘ |
| appearsInWork | Arithmetica Infinitorum ⓘ |
| category |
Formulae involving π
ⓘ
Infinite products ⓘ |
| convergenceType | slow convergence ⓘ |
| convergesTo | π/2 ⓘ |
| derivationMethod |
integration of powers of cosine
ⓘ
integration of powers of sine ⓘ use of recursion formulas for ∫₀^{π/2} cos^n x dx ⓘ use of recursion formulas for ∫₀^{π/2} sin^n x dx ⓘ |
| discoveredBy | John Wallis ⓘ |
| field |
calculus
ⓘ
mathematical analysis ⓘ number theory ⓘ |
| hasLimitForm | lim_{n→∞} ∏_{k=1}^{n} (4k^2)/(4k^2 - 1) = π/2 ⓘ |
| historicalSignificance |
contributed to development of analysis
ⓘ
early example of rigorous use of infinite products ⓘ influenced later work on infinite series and products ⓘ |
| isOneOf | earliest infinite product representations of π ⓘ |
| mainExpression | π/2 = ∏_{n=1}^{∞} (4n^2)/(4n^2 - 1) ⓘ |
| namedAfter | John Wallis ⓘ |
| namedEntity | yes ⓘ |
| publicationCentury | 17th century ⓘ |
| relatedConstant |
π
ⓘ
π/2 ⓘ |
| relatedTo |
Beta function
ⓘ
Euler’s infinite product for sine ⓘ
linked to:
Euler’s identity for sine product
Gamma function ⓘ Stirling’s approximation ⓘ
linked to:
Stirling's approximation
cosine function ⓘ sine function ⓘ |
| termPattern |
factors of the form (4n^2)/(4n^2 - 1)
ⓘ
ratio of even to odd integers ⓘ |
| usedFor |
approximating π
ⓘ
proving properties of trigonometric integrals ⓘ studying infinite products ⓘ |
| yearApproximate | 1655 ⓘ |
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: Wallis product Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.