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Fungrim entry: b6017f

logG ⁣(1z)=logG ⁣(1+z)log ⁣(2π)z+0zπxcot ⁣(πx)dx\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:zC  and  z(,1][1,)z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, -1\right] \cup \left[1, \infty\right)
\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)
Fungrim symbol Notation Short description
LogBarnesGlogG(z)\log G(z) Logarithmic Barnes G-function
Loglog(z)\log(z) Natural logarithm
Piπ\pi The constant pi (3.14...)
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
Source code for this entry:
    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))))),
    Assumptions(And(Element(z, CC), NotElement(z, Union(OpenClosedInterval(Neg(Infinity), -1), ClosedOpenInterval(1, Infinity))))))

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2021-03-15 19:12:00.328586 UTC