# Fungrim entry: 645c98

$R_{N}\!\left(z\right) = \frac{1}{2 N \left(2 N + 1\right)} \int_{0}^{\infty} \frac{B_{2 N + 1}\!\left(t - \left\lfloor t \right\rfloor\right)}{{\left(t + z\right)}^{2 N}} \, dt$
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}{2 N \left(2 N + 1\right)} \int_{0}^{\infty} \frac{B_{2 N + 1}\!\left(t - \left\lfloor t \right\rfloor\right)}{{\left(t + z\right)}^{2 N}} \, 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
Integral$\int_{a}^{b} f(x) \, dx$ Integral
BernoulliPolynomial$B_{n}\!\left(z\right)$ Bernoulli polynomial
Pow${a}^{b}$ Power
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("645c98"),
Formula(Equal(LogBarnesGRemainder(N, z), Mul(Div(1, Mul(Mul(2, N), Add(Mul(2, N), 1))), Integral(Div(BernoulliPolynomial(Add(Mul(2, N), 1), Sub(t, Floor(t))), Pow(Add(t, z), Mul(2, N))), 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