# Fungrim entry: 52ea5f

$\operatorname{Li}_{s}\!\left(z\right) = \frac{\Gamma\!\left(1 - s\right)}{{\left(2 \pi\right)}^{1 - s}} \left({i}^{1 - s} \zeta\!\left(1 - s, \frac{1}{2} + \frac{\log\!\left(-z\right)}{2 \pi i}\right) + {i}^{s - 1} \zeta\!\left(1 - s, \frac{1}{2} - \frac{\log\!\left(-z\right)}{2 \pi i}\right)\right)$
Assumptions:$s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left\{0, 1\right\} \;\mathbin{\operatorname{and}}\; s \notin \mathbb{Z}_{\ge 0}$
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$\Gamma(z)$ Gamma function
Pow${a}^{b}$ Power
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
HurwitzZeta$\zeta\!\left(s, a\right)$ Hurwitz zeta function
Log$\log(z)$ Natural logarithm
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ 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)))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC