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ID: INPUT{id.yaml}

Title: >
  Values of the Barnes $G$-function at rational numbers

Definition: >
  The list contains values of the Barnes $G$-function 
  at rational number $s = a/b$.
  
  For a complex number $s$, $G(s)$ can be defined via the Weierstrass product
  $G(s+1) = (2\pi)^{s/2} \exp\big(-\frac{s+s^2(1+\gamma)}{2}\big)
  \prod_{k=1}^\infty (1+\frac{s}{k})^k \exp\big(\frac{s^2}{2k}-s\big)$,
  where $\gamma$ is the Euler-Mascheroni constant.

Parameters:
  s:
    type: Q
         
Comments:
  
Formulas: 
  formula-functional-equation: >
    Functional equation: 
    $G(s+1) = \Gamma(s)G(s)$ 
    for all $s\in\mathbb{C} \backslash \mathbb{Z}_{\leq 0}$,
    where $\Gamma(s)$ is the Gamma function.
  formula-factorial: >
    $G(n) = 0$ for integers $n \leq 0$,
    and $G(n) = \prod_{k=0}^{n-2} k!$ for integers $n\geq 1$.
  formula-Kinkelin-reflection: >
    Kinkelin's reflection formula:
    $\log G(1-s) = \log G(1+s) - s\log 2\pi + \int_0^s \pi x \cot \pi x \, dx$.
  formula_G_at_1_over_2: >
    $G(1/2) = 2^{1/24} \exp(\frac{3}{2}\zeta'(-1)) \pi^{-1/4}$.
  
Programs: 
  program-sage:
    language: Sage
    code: |
      numbers = {a/b: mpmath.barnesg(RIF(a/b)) 
                      for b,a in cartesian_product(([1..30],[-30..30])) 
                      if gcd(a,b) == 1}

References:

Links:
  Wiki:
    title: "Wikipedia: Barnes G-function"
    url: https://en.wikipedia.org/wiki/Barnes_G-function

Similar tables:
  
Keywords:
  
Tags:
- special values

Data properties:
  type: R
  complete: no
  
Display properties:
  number-header: $G(s)$

Numbers: INPUT{numbers.yaml}

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Values of the Barnes $G$-function at rational numbers

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Numbers

$s$

$G(s)$

INPUT{numbers.yaml} (not shown in preview)

Definition

The list contains values of the Barnes $G$-function at rational number $s = a/b$.
For a complex number $s$, $G(s)$ can be defined via the Weierstrass product $G(s+1) = (2\pi)^{s/2} \exp\big(-\frac{s+s^2(1+\gamma)}{2}\big) \prod_{k=1}^\infty (1+\frac{s}{k})^k \exp\big(\frac{s^2}{2k}-s\big)$, where $\gamma$ is the Euler-Mascheroni constant.

Parameters

$s$

— rational number

Formulas

(1)

Functional equation: $G(s+1) = \Gamma(s)G(s)$ for all $s\in\mathbb{C} \backslash \mathbb{Z}_{\leq 0}$, where $\Gamma(s)$ is the Gamma function.

(2)

$G(n) = 0$ for integers $n \leq 0$, and $G(n) = \prod_{k=0}^{n-2} k!$ for integers $n\geq 1$.

(3)

Kinkelin's reflection formula: $\log G(1-s) = \log G(1+s) - s\log 2\pi + \int_0^s \pi x \cot \pi x \, dx$.

(4)

$G(1/2) = 2^{1/24} \exp(\frac{3}{2}\zeta'(-1)) \pi^{-1/4}$.

Programs

(P1)

Sage

numbers = {a/b: mpmath.barnesg(RIF(a/b)) 
                for b,a in cartesian_product(([1..30],[-30..30])) 
                if gcd(a,b) == 1}

Links

[1]

Wikipedia: Barnes G-function

Data properties

Entries are of type: real number

Table is complete: no