Fungrim entry: 092cee

R_{N}\!\left(z\right) = \frac{1}{{z}^{2 N}} \frac{{\left(-1\right)}^{N + 1}}{\pi} \int_{0}^{\infty} \frac{{t}^{2 N - 1}}{1 + {\left(\frac{t}{z}\right)}^{2}} \operatorname{Li}_{2}\!\left({e}^{-2 \pi t}\right) \, dt

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

References:

https://dx.doi.org/10.1098/rspa.2014.0534

TeX:

R_{N}\!\left(z\right) = \frac{1}{{z}^{2 N}} \frac{{\left(-1\right)}^{N + 1}}{\pi} \int_{0}^{\infty} \frac{{t}^{2 N - 1}}{1 + {\left(\frac{t}{z}\right)}^{2}} \operatorname{Li}_{2}\!\left({e}^{-2 \pi t}\right) \, dt

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesGRemainder`	$R_{N}\!\left(z\right)$	Remainder term in asymptotic expansion of logarithmic Barnes G-function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("092cee"),
    Formula(Equal(LogBarnesGRemainder(N, z), Mul(Mul(Div(1, Pow(z, Mul(2, N))), Div(Pow(-1, Add(N, 1)), Pi)), Integral(Mul(Div(Pow(t, Sub(Mul(2, N), 1)), Add(1, Pow(Div(t, z), 2))), PolyLog(2, Exp(Neg(Mul(Mul(2, Pi), t))))), For(t, 0, Infinity))))),
    Variables(z, N),
    Assumptions(And(Element(z, CC), Greater(Re(z), 0), Element(N, ZZGreaterEqual(1)))),
    References("https://dx.doi.org/10.1098/rspa.2014.0534"))

Topics using this entry

Barnes G-function