# Fungrim entry: 54d4e2

$G\!\left(z + 1\right) = {\left(2 \pi\right)}^{z / 2} {e}^{-\left( z + \left(\gamma + 1\right) {z}^{2} \right) / 2} \prod_{k=1}^{\infty} \left[{\left(1 + \frac{z}{k}\right)}^{k} \exp\!\left(\frac{{z}^{2}}{2 k} - z\right)\right]$
Assumptions:$z \in \mathbb{C}$
TeX:
G\!\left(z + 1\right) = {\left(2 \pi\right)}^{z / 2} {e}^{-\left( z + \left(\gamma + 1\right) {z}^{2} \right) / 2} \prod_{k=1}^{\infty} \left[{\left(1 + \frac{z}{k}\right)}^{k} \exp\!\left(\frac{{z}^{2}}{2 k} - z\right)\right]

z \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
BarnesG$G(z)$ Barnes G-function
Pow${a}^{b}$ Power
Pi$\pi$ The constant pi (3.14...)
Exp${e}^{z}$ Exponential function
ConstGamma$\gamma$ The constant gamma (0.577...)
Product$\prod_{n} f(n)$ Product
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("54d4e2"),
Formula(Equal(BarnesG(Add(z, 1)), Mul(Mul(Pow(Mul(2, Pi), Div(z, 2)), Exp(Neg(Div(Add(z, Mul(Add(ConstGamma, 1), Pow(z, 2))), 2)))), Product(Brackets(Mul(Pow(Add(1, Div(z, k)), k), Exp(Sub(Div(Pow(z, 2), Mul(2, k)), z)))), For(k, 1, Infinity))))),
Variables(z),
Assumptions(Element(z, CC)))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC