Assumptions:
TeX:
\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}
Definitions:
Fungrim symbol | Notation | Short description |
---|---|---|
Gamma | Gamma function | |
Pow | Power | |
Pi | The constant pi (3.14...) | |
ConstI | Imaginary unit | |
HurwitzZeta | Hurwitz zeta function | |
Log | Natural logarithm | |
CC | Complex numbers | |
ZZGreaterEqual | Integers greater than or equal to n |
Source code for this entry:
Entry(ID("52ea5f"), 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)))))