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

Barnes G-function