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Fungrim entry: 645c98

RN ⁣(z)=12N(2N+1)0B2N+1 ⁣(tt)(t+z)2NdtR_{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:zC  and  Re(z)>0  and  NZ1z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}
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}
Fungrim symbol Notation Short description
LogBarnesGRemainderRN ⁣(z)R_{N}\!\left(z\right) Remainder term in asymptotic expansion of logarithmic Barnes G-function
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
BernoulliPolynomialBn ⁣(z)B_{n}\!\left(z\right) Bernoulli polynomial
Powab{a}^{b} Power
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    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)))),

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2021-03-15 19:12:00.328586 UTC