Fungrim entry: 82b410

\log G\!\left(1 - z\right) = \log G\!\left(1 + z\right) + \begin{cases} F(z), & 0 < \operatorname{Re}(z) < 1 \;\mathbin{\operatorname{or}}\; \operatorname{Im}(z) > 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(z) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) < 1\right)\\-F\!\left(-z\right), & -1 < \operatorname{Re}(z) < 0 \;\mathbin{\operatorname{or}}\; \operatorname{Im}(z) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(z) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > -1\right)\\ \end{cases}\; \text{ where } F(z) = \frac{\pi i}{2} \left({z}^{2} - z + \frac{1}{6}\right) - z \left(\log \Gamma(z) + \log \Gamma\!\left(1 - z\right)\right) - \frac{i}{2 \pi} \operatorname{Li}_{2}\!\left({e}^{2 \pi i z}\right)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \mathbb{Z}

TeX:

\log G\!\left(1 - z\right) = \log G\!\left(1 + z\right) + \begin{cases} F(z), & 0 < \operatorname{Re}(z) < 1 \;\mathbin{\operatorname{or}}\; \operatorname{Im}(z) > 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(z) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) < 1\right)\\-F\!\left(-z\right), & -1 < \operatorname{Re}(z) < 0 \;\mathbin{\operatorname{or}}\; \operatorname{Im}(z) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(z) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > -1\right)\\ \end{cases}\; \text{ where } F(z) = \frac{\pi i}{2} \left({z}^{2} - z + \frac{1}{6}\right) - z \left(\log \Gamma(z) + \log \Gamma\!\left(1 - z\right)\right) - \frac{i}{2 \pi} \operatorname{Li}_{2}\!\left({e}^{2 \pi i z}\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Re`	$\operatorname{Re}(z)$	Real part
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("82b410"),
    Formula(Equal(LogBarnesG(Sub(1, z)), Add(LogBarnesG(Add(1, z)), Where(Cases(Tuple(F(z), Or(Less(0, Re(z), 1), Greater(Im(z), 0), And(Equal(Im(z), 0), Less(Re(z), 1)))), Tuple(Neg(F(Neg(z))), Or(Less(-1, Re(z), 0), Less(Im(z), 0), And(Equal(Im(z), 0), Greater(Re(z), -1))))), Equal(F(z), Sub(Sub(Mul(Div(Mul(Pi, ConstI), 2), Add(Sub(Pow(z, 2), z), Div(1, 6))), Mul(z, Add(LogGamma(z), LogGamma(Sub(1, z))))), Mul(Div(ConstI, Mul(2, Pi)), PolyLog(2, Exp(Mul(Mul(Mul(2, Pi), ConstI), z)))))))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, ZZ))))

Topics using this entry

Barnes G-function