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

Barnes G-function