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Fungrim entry: 54d4e2

G ⁣(z+1)=(2π)z/2e(z+(γ+1)z2)/2k=1[(1+zk)kexp ⁣(z22kz)]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:zCz \in \mathbb{C}
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}
Fungrim symbol Notation Short description
BarnesGG(z)G(z) Barnes G-function
Powab{a}^{b} Power
Piπ\pi The constant pi (3.14...)
Expez{e}^{z} Exponential function
ConstGammaγ\gamma The constant gamma (0.577...)
Productnf(n)\prod_{n} f(n) Product
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
Source code for this entry:
    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))))),
    Assumptions(Element(z, CC)))

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2021-03-15 19:12:00.328586 UTC