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

Hurwitz zeta function