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

Barnes G-function