Fungrim entry: b64782

\log G\!\left(z + 1\right) = \frac{{z}^{2}}{4} \left(2 \log(z) - 3\right) + \frac{z \log\!\left(2 \pi\right)}{2} + \frac{1}{12} - \log(A) - \int_{0}^{\infty} \frac{x \log\!\left({x}^{2} + {z}^{2}\right)}{{e}^{2 \pi x} - 1} \, 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) = \frac{{z}^{2}}{4} \left(2 \log(z) - 3\right) + \frac{z \log\!\left(2 \pi\right)}{2} + \frac{1}{12} - \log(A) - \int_{0}^{\infty} \frac{x \log\!\left({x}^{2} + {z}^{2}\right)}{{e}^{2 \pi x} - 1} \, 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
`Pow`	${a}^{b}$	Power
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`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("b64782"),
    Formula(Equal(LogBarnesG(Add(z, 1)), Sub(Sub(Add(Add(Mul(Div(Pow(z, 2), 4), Sub(Mul(2, Log(z)), 3)), Div(Mul(z, Log(Mul(2, Pi))), 2)), Div(1, 12)), Log(ConstGlaisher)), Integral(Div(Mul(x, Log(Add(Pow(x, 2), Pow(z, 2)))), Sub(Exp(Mul(Mul(2, Pi), x)), 1)), 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