# 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC