# Fungrim entry: d1a0ec

$\log G\!\left(1 - x\right) = \log G\!\left(1 + x\right) + x \log\!\left(\frac{\left|\sin\!\left(\pi x\right)\right|}{\pi}\right) + \frac{1}{2 \pi} \operatorname{Im}\!\left(\operatorname{Li}_{2}\!\left({e}^{2 \pi i x}\right)\right) + \operatorname{sgn}(x) n \left(n + 1\right) \frac{\pi i}{2}\; \text{ where } n = \left\lfloor x \right\rfloor$
Assumptions:$x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \notin \mathbb{Z}$
TeX:
\log G\!\left(1 - x\right) = \log G\!\left(1 + x\right) + x \log\!\left(\frac{\left|\sin\!\left(\pi x\right)\right|}{\pi}\right) + \frac{1}{2 \pi} \operatorname{Im}\!\left(\operatorname{Li}_{2}\!\left({e}^{2 \pi i x}\right)\right) + \operatorname{sgn}(x) n \left(n + 1\right) \frac{\pi i}{2}\; \text{ where } n = \left\lfloor x \right\rfloor

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \notin \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
LogBarnesG$\log G(z)$ Logarithmic Barnes G-function
Log$\log(z)$ Natural logarithm
Abs$\left|z\right|$ Absolute value
Sin$\sin(z)$ Sine
Pi$\pi$ The constant pi (3.14...)
Im$\operatorname{Im}(z)$ Imaginary part
Exp${e}^{z}$ Exponential function
ConstI$i$ Imaginary unit
Sign$\operatorname{sgn}(z)$ Sign function
RR$\mathbb{R}$ Real numbers
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("d1a0ec"),
Variables(x),
Assumptions(And(Element(x, RR), NotElement(x, ZZ))))

## 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