# Fungrim entry: 541e2e

$G\!\left(1 - x\right) = {\left(-1\right)}^{\left\lfloor \left( x - 1 \right) / 2 \right\rfloor + 1} G\!\left(1 + x\right) {\left(\frac{\left|\sin\!\left(\pi x\right)\right|}{\pi}\right)}^{x} \exp\!\left(\frac{1}{2 \pi} \operatorname{Im}\!\left(\operatorname{Li}_{2}\!\left({e}^{2 \pi i x}\right)\right)\right)$
Assumptions:$x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \notin \{-1, -2, \ldots\}$
References:
• https://doi.org/10.1145/384101.384104
TeX:
G\!\left(1 - x\right) = {\left(-1\right)}^{\left\lfloor \left( x - 1 \right) / 2 \right\rfloor + 1} G\!\left(1 + x\right) {\left(\frac{\left|\sin\!\left(\pi x\right)\right|}{\pi}\right)}^{x} \exp\!\left(\frac{1}{2 \pi} \operatorname{Im}\!\left(\operatorname{Li}_{2}\!\left({e}^{2 \pi i x}\right)\right)\right)

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \notin \{-1, -2, \ldots\}
Definitions:
Fungrim symbol Notation Short description
BarnesG$G(z)$ Barnes G-function
Pow${a}^{b}$ Power
Abs$\left|z\right|$ Absolute value
Sin$\sin(z)$ Sine
Pi$\pi$ The constant pi (3.14...)
Exp${e}^{z}$ Exponential function
Im$\operatorname{Im}(z)$ Imaginary part
ConstI$i$ Imaginary unit
RR$\mathbb{R}$ Real numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
Source code for this entry:
Entry(ID("541e2e"),
Formula(Equal(BarnesG(Sub(1, x)), Mul(Mul(Mul(Pow(-1, Add(Floor(Div(Sub(x, 1), 2)), 1)), BarnesG(Add(1, x))), Pow(Div(Abs(Sin(Mul(Pi, x))), Pi), x)), Exp(Mul(Div(1, Mul(2, Pi)), Im(PolyLog(2, Exp(Mul(Mul(Mul(2, Pi), ConstI), x))))))))),
Variables(x),
Assumptions(And(Element(x, RR), NotElement(x, ZZLessEqual(-1)))),
References("https://doi.org/10.1145/384101.384104"))

## 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