# 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

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