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Fungrim entry: 52ea5f

Lis ⁣(z)=Γ ⁣(1s)(2π)1s(i1sζ ⁣(1s,12+log ⁣(z)2πi)+is1ζ ⁣(1s,12log ⁣(z)2πi))\operatorname{Li}_{s}\!\left(z\right) = \frac{\Gamma\!\left(1 - s\right)}{{\left(2 \pi\right)}^{1 - s}} \left({i}^{1 - s} \zeta\!\left(1 - s, \frac{1}{2} + \frac{\log\!\left(-z\right)}{2 \pi i}\right) + {i}^{s - 1} \zeta\!\left(1 - s, \frac{1}{2} - \frac{\log\!\left(-z\right)}{2 \pi i}\right)\right)
Assumptions:sC  and  zC  and  z{0,1}  and  sZ0s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left\{0, 1\right\} \;\mathbin{\operatorname{and}}\; s \notin \mathbb{Z}_{\ge 0}
\operatorname{Li}_{s}\!\left(z\right) = \frac{\Gamma\!\left(1 - s\right)}{{\left(2 \pi\right)}^{1 - s}} \left({i}^{1 - s} \zeta\!\left(1 - s, \frac{1}{2} + \frac{\log\!\left(-z\right)}{2 \pi i}\right) + {i}^{s - 1} \zeta\!\left(1 - s, \frac{1}{2} - \frac{\log\!\left(-z\right)}{2 \pi i}\right)\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left\{0, 1\right\} \;\mathbin{\operatorname{and}}\; s \notin \mathbb{Z}_{\ge 0}
Fungrim symbol Notation Short description
GammaΓ(z)\Gamma(z) Gamma function
Powab{a}^{b} Power
Piπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
Loglog(z)\log(z) Natural logarithm
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(Equal(PolyLog(s, z), Mul(Div(Gamma(Sub(1, s)), Pow(Mul(2, Pi), Sub(1, s))), Add(Mul(Pow(ConstI, Sub(1, s)), HurwitzZeta(Sub(1, s), Add(Div(1, 2), Div(Log(Neg(z)), Mul(Mul(2, Pi), ConstI))))), Mul(Pow(ConstI, Sub(s, 1)), HurwitzZeta(Sub(1, s), Sub(Div(1, 2), Div(Log(Neg(z)), Mul(Mul(2, Pi), ConstI))))))))),
    Variables(s, z),
    Assumptions(And(Element(s, CC), Element(z, CC), NotElement(z, Set(0, 1)), NotElement(s, ZZGreaterEqual(0)))))

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2021-03-15 19:12:00.328586 UTC