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

Barnes G-function