Barnes G-function
E865106
The Barnes G-function is a special function that generalizes the superfactorial and is closely related to the gamma and multiple gamma functions in complex analysis and number theory.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
complex-analytic function
ⓘ
entire function ⓘ special function ⓘ |
| appearsIn |
evaluation of certain infinite products
ⓘ
multiple zeta and multiple gamma theory ⓘ |
| category | Barnes functions NERFINISHED ⓘ |
| domain | complex plane ⓘ |
| field |
complex analysis
ⓘ
number theory ⓘ |
| generalizes |
factorial-related products
ⓘ
superfactorial ⓘ |
| growthType | order 2 entire function ⓘ |
| hasAsymptoticExpansion | log G(z+1) expressed using Bernoulli numbers ⓘ |
| hasLogDerivative | log G(z+1)= (z/2)log(2π) - (3z^2/4) + O(z log z) as z→∞ (up to constants) ⓘ |
| hasProperty | log-convex on positive real axis ⓘ |
| hasWeierstrassProduct | infinite product representation involving Γ(z) ⓘ |
| hasZeroAt |
z=-1
ⓘ
z=-2 ⓘ z=-3 ⓘ z=0 ⓘ |
| introducedBy | Ernest William Barnes NERFINISHED ⓘ |
| introducedIn | early 20th century ⓘ |
| isEntireIn | z ⓘ |
| isSolutionOf | functional equation of second order in shifts ⓘ |
| namedAfter | Ernest William Barnes NERFINISHED ⓘ |
| normalization | G(1)=1 ⓘ |
| notation | G(z) ⓘ |
| relatedTo |
Barnes multiple zeta function
NERFINISHED
ⓘ
K-function of Kinkelin NERFINISHED ⓘ Riemann zeta function NERFINISHED ⓘ gamma function ⓘ multiple gamma function ⓘ superfactorial function n$!$ (product of factorials) ⓘ |
| satisfies | G(z)≠0 for non-integer complex z ⓘ |
| satisfiesFunctionalEquation | G(z+1)=Γ(z)G(z) ⓘ |
| satisfiesRecurrence | G(n+1)=Γ(n)G(n) for positive integers n ⓘ |
| specialValue | G(n+1)=∏_{k=1}^{n-1}k! for positive integers n ⓘ |
| usedFor |
evaluation of partition functions in physics
ⓘ
regularization of infinite products ⓘ |
| usedIn |
asymptotic analysis
ⓘ
determinants of Laplacians ⓘ quantum field theory ⓘ random matrix theory ⓘ spectral theory ⓘ |
| valueAt |
G(1)=1
ⓘ
G(2)=1 ⓘ G(3)=1 ⓘ |
| zeroSet | non-positive integers ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.