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Fungrim entry: 53a2a1

Rn ⁣(z)=0B2nB2n ⁣(tt)2n(z+t)2ndtR_{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:zCandz(,0]andnZ1z \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
StirlingSeriesRemainderRn ⁣(z)R_{n}\!\left(z\right) Remainder term in the Stirling series for the logarithmic gamma function
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
BernoulliBBnB_{n} Bernoulli number
BernoulliPolynomialBn ⁣(z)B_{n}\!\left(z\right) Bernoulli polynomial
Powab{a}^{b} Power
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
ZZGreaterEqualZn\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)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC