Fungrim entry: 23ed69

\log G\!\left(1 - z\right) = \log G\!\left(1 + z\right) - \log\!\left(2 \pi\right) z + \begin{cases} \int_{0}^{i} \pi x \cot\!\left(\pi x\right) \, dx + \int_{i}^{z} \pi x \cot\!\left(\pi x\right) \, dx, & -1 < \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)\\\int_{0}^{-i} \pi x \cot\!\left(\pi x\right) \, dx + \int_{-i}^{z} \pi x \cot\!\left(\pi x\right) \, dx, & -1 < \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)\\ \end{cases}

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) - \log\!\left(2 \pi\right) z + \begin{cases} \int_{0}^{i} \pi x \cot\!\left(\pi x\right) \, dx + \int_{i}^{z} \pi x \cot\!\left(\pi x\right) \, dx, & -1 < \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)\\\int_{0}^{-i} \pi x \cot\!\left(\pi x\right) \, dx + \int_{-i}^{z} \pi x \cot\!\left(\pi x\right) \, dx, & -1 < \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)\\ \end{cases}

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
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`ConstI`	$i$	Imaginary unit
`Re`	$\operatorname{Re}(z)$	Real part
`Im`	$\operatorname{Im}(z)$	Imaginary part
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("23ed69"),
    Formula(Equal(LogBarnesG(Sub(1, z)), Add(Sub(LogBarnesG(Add(1, z)), Mul(Log(Mul(2, Pi)), z)), Cases(Tuple(Add(Integral(Mul(Mul(Pi, x), Cot(Mul(Pi, x))), For(x, 0, ConstI)), Integral(Mul(Mul(Pi, x), Cot(Mul(Pi, x))), For(x, ConstI, z))), Or(Less(-1, Re(z), 1), Greater(Im(z), 0), And(Equal(Im(z), 0), Less(Re(z), 1)))), Tuple(Add(Integral(Mul(Mul(Pi, x), Cot(Mul(Pi, x))), For(x, 0, Neg(ConstI))), Integral(Mul(Mul(Pi, x), Cot(Mul(Pi, x))), For(x, Neg(ConstI), z))), Or(Less(-1, Re(z), 1), Less(Im(z), 0), And(Equal(Im(z), 0), Greater(Re(z), -1)))))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, ZZ))))

Topics using this entry

Barnes G-function