Gauss multiplication formula
E596521
The Gauss multiplication formula is a classical identity in complex analysis that expresses the gamma function of a multiple of a variable as a product of gamma functions evaluated at shifted fractions of that variable.
Statements (39)
| Predicate | Object |
|---|---|
| instanceOf |
formula involving the gamma function
ⓘ
identity in complex analysis ⓘ mathematical formula ⓘ |
| appearsIn |
textbooks on complex analysis
ⓘ
textbooks on special functions ⓘ treatises on the gamma function ⓘ |
| category |
gamma function identities
ⓘ
multiplication theorems in analysis ⓘ |
| domainVariable | complex variable z ⓘ |
| expresses | gamma function of a multiple of a variable as a product of shifted gamma functions ⓘ |
| field |
complex analysis
ⓘ
mathematical analysis ⓘ special functions ⓘ |
| givesExpressionFor | Γ(nz) ⓘ |
| hasProperty |
extends to meromorphic identity on ℂ
ⓘ
provides finite product representation for Γ(nz) in terms of Γ at shifted arguments ⓘ |
| involvesFunction | gamma function ⓘ |
| isGeneralizationOf |
duplication formula for the gamma function
ⓘ
triplication formula for the gamma function ⓘ |
| namedAfter | Carl Friedrich Gauss NERFINISHED ⓘ |
| parameter | positive integer n ⓘ |
| relatedTo |
Euler reflection formula
NERFINISHED
ⓘ
Weierstrass product for the gamma function NERFINISHED ⓘ |
| relates |
Γ(nz)
ⓘ
Γ(z) ⓘ Γ(z+(n-1)/n) ⓘ Γ(z+1/n) ⓘ Γ(z+2/n) ⓘ |
| requiresCondition | n is a positive integer for the standard form ⓘ |
| usedIn |
analytic number theory
ⓘ
asymptotic analysis of special functions ⓘ derivations involving the beta function ⓘ evaluation of certain infinite products ⓘ functional equations for special functions ⓘ |
| usesConstant | (2π)^{(1-n)/2} ⓘ |
| usesFactor | n^{nz-1/2} ⓘ |
| usesProduct | ∏_{k=0}^{n-1} Γ(z + k/n) ⓘ |
| validFor |
n ∈ ℕ, n ≥ 1
ⓘ
z ∈ ℂ with Γ defined and finite ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.