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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.