# Fungrim entry: b6017f

$\log G\!\left(1 - z\right) = \log G\!\left(1 + z\right) - \log\!\left(2 \pi\right) z + \int_{0}^{z} \pi x \cot\!\left(\pi x\right) \, dx$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, -1\right] \cup \left[1, \infty\right)$
TeX:
\log G\!\left(1 - z\right) = \log G\!\left(1 + z\right) - \log\!\left(2 \pi\right) z + \int_{0}^{z} \pi x \cot\!\left(\pi x\right) \, dx

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, -1\right] \cup \left[1, \infty\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...)
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
ClosedOpenInterval$\left[a, b\right)$ Closed-open interval
Source code for this entry:
Entry(ID("b6017f"),
Formula(Equal(LogBarnesG(Sub(1, z)), Add(Sub(LogBarnesG(Add(1, z)), Mul(Log(Mul(2, Pi)), z)), Integral(Mul(Mul(Pi, x), Cot(Mul(Pi, x))), For(x, 0, z))))),
Variables(z),
Assumptions(And(Element(z, CC), NotElement(z, Union(OpenClosedInterval(Neg(Infinity), -1), ClosedOpenInterval(1, Infinity))))))

## 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