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.
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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| namedEntity | yes ⓘ |
| publicationCentury | 17th century ⓘ |
| relatedConstant |
π
ⓘ
π/2 ⓘ |
| relatedTo |
Beta function
ⓘ
Euler’s infinite product for sine NERFINISHED ⓘ Gamma function NERFINISHED ⓘ Stirling’s approximation NERFINISHED ⓘ 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.