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"),
    Formula(Equal(LogBarnesG(Sub(1, x)), Add(Add(Add(LogBarnesG(Add(1, x)), Mul(x, Log(Div(Abs(Sin(Mul(Pi, x))), Pi)))), Mul(Div(1, Mul(2, Pi)), Im(PolyLog(2, Exp(Mul(Mul(Mul(2, Pi), ConstI), x)))))), Where(Mul(Mul(Sign(x), Mul(n, Add(n, 1))), Div(Mul(Pi, ConstI), 2)), Equal(n, Floor(x)))))),
    Variables(x),
    Assumptions(And(Element(x, RR), NotElement(x, ZZ))))

Topics using this entry

Barnes G-function