# Fungrim entry: 53a2a1

$R_{n}\!\left(z\right) = \int_{0}^{\infty} \frac{B_{2 n} - B_{2 n}\!\left(t - \left\lfloor t \right\rfloor\right)}{2 n {\left(z + t\right)}^{2 n}} \, dt$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}$
TeX:
R_{n}\!\left(z\right) = \int_{0}^{\infty} \frac{B_{2 n} - B_{2 n}\!\left(t - \left\lfloor t \right\rfloor\right)}{2 n {\left(z + t\right)}^{2 n}} \, dt

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
StirlingSeriesRemainder$R_{n}\!\left(z\right)$ Remainder term in the Stirling series for the logarithmic gamma function
Integral$\int_{a}^{b} f(x) \, dx$ Integral
BernoulliB$B_{n}$ Bernoulli number
BernoulliPolynomial$B_{n}\!\left(z\right)$ Bernoulli polynomial
Pow${a}^{b}$ Power
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("53a2a1"),
Formula(Equal(StirlingSeriesRemainder(n, z), Integral(Div(Sub(BernoulliB(Mul(2, n)), BernoulliPolynomial(Mul(2, n), Sub(t, Floor(t)))), Mul(Mul(2, n), Pow(Add(z, t), Mul(2, n)))), For(t, 0, Infinity)))),
Variables(z, n),
Assumptions(And(Element(z, CC), NotElement(z, OpenClosedInterval(Neg(Infinity), 0)), Element(n, ZZGreaterEqual(1)))))

## 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