Fungrim entry: 6395ee

\log G\!\left(z + 1\right) = z \log \Gamma(z) + \frac{{z}^{2}}{4} - \frac{\log(z)}{2} B_{2}\!\left(z\right) - \log(A) - \int_{0}^{\infty} \frac{{e}^{-z x}}{{x}^{2}} \left(\frac{1}{1 - {e}^{-x}} - \frac{1}{x} - \frac{1}{2} - \frac{x}{12}\right) \, dx

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0

References:

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

TeX:

\log G\!\left(z + 1\right) = z \log \Gamma(z) + \frac{{z}^{2}}{4} - \frac{\log(z)}{2} B_{2}\!\left(z\right) - \log(A) - \int_{0}^{\infty} \frac{{e}^{-z x}}{{x}^{2}} \left(\frac{1}{1 - {e}^{-x}} - \frac{1}{x} - \frac{1}{2} - \frac{x}{12}\right) \, dx

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`Pow`	${a}^{b}$	Power
`Log`	$\log(z)$	Natural logarithm
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("6395ee"),
    Formula(Equal(LogBarnesG(Add(z, 1)), Sub(Sub(Sub(Add(Mul(z, LogGamma(z)), Div(Pow(z, 2), 4)), Mul(Div(Log(z), 2), BernoulliPolynomial(2, z))), Log(ConstGlaisher)), Integral(Mul(Div(Exp(Mul(Neg(z), x)), Pow(x, 2)), Sub(Sub(Sub(Div(1, Sub(1, Exp(Neg(x)))), Div(1, x)), Div(1, 2)), Div(x, 12))), For(x, 0, Infinity))))),
    Variables(z),
    Assumptions(And(Element(z, CC), Greater(Re(z), 0))),
    References("https://arxiv.org/abs/math/0308086"))

Topics using this entry

Barnes G-function