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Fungrim entry: 23ed69

logG ⁣(1z)=logG ⁣(1+z)log ⁣(2π)z+{0iπxcot ⁣(πx)dx+izπxcot ⁣(πx)dx,1<Re(z)<1  or  Im(z)>0  or  (Im(z)=0  and  Re(z)<1)0iπxcot ⁣(πx)dx+izπxcot ⁣(πx)dx,1<Re(z)<1  or  Im(z)<0  or  (Im(z)=0  and  Re(z)>1)\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:zC  and  zZz \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
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
ConstIii Imaginary unit
ReRe(z)\operatorname{Re}(z) Real part
ImIm(z)\operatorname{Im}(z) Imaginary part
CCC\mathbb{C} Complex numbers
ZZZ\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))))

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2021-03-15 19:12:00.328586 UTC