Fungrim entry: 8c96a5

\log G\!\left(z + 1\right) = \frac{z \left(1 - z\right)}{2} + \frac{z}{2} \log\!\left(2 \pi\right) + z \log \Gamma(z) - \int_{0}^{z} \log \Gamma(x) \, dx

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, -1\right]

References:

https://arxiv.org/abs/math/0308086

TeX:

\log G\!\left(z + 1\right) = \frac{z \left(1 - z\right)}{2} + \frac{z}{2} \log\!\left(2 \pi\right) + z \log \Gamma(z) - \int_{0}^{z} \log \Gamma(x) \, dx

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, -1\right]

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("8c96a5"),
    Formula(Equal(LogBarnesG(Add(z, 1)), Sub(Add(Add(Div(Mul(z, Sub(1, z)), 2), Mul(Div(z, 2), Log(Mul(2, Pi)))), Mul(z, LogGamma(z))), Integral(LogGamma(x), For(x, 0, z))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, OpenClosedInterval(Neg(Infinity), -1)))),
    References("https://arxiv.org/abs/math/0308086"))

Topics using this entry

Barnes G-function