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
`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)))), Tuple(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

Gamma function