# 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

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