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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Gamma function reflection formula | 2 |
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
NERFINISHED
ⓘ
sine function ⓘ |
| namedAfter | Leonhard Euler NERFINISHED ⓘ |
| relatedTo |
Euler’s formula for the sine function via infinite product
NERFINISHED
ⓘ
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 ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Gamma function reflection formula
this entity surface form:
Gamma function reflection formula