Fungrim home page

Fungrim entry: b16d00

RN ⁣(z)=0(t2coth ⁣(t2)k=0NB2k(2k)!t2k)eztt3dtR_{N}\!\left(z\right) = \int_{0}^{\infty} \left(\frac{t}{2} \coth\!\left(\frac{t}{2}\right) - \sum_{k=0}^{N} \frac{B_{2 k}}{\left(2 k\right)!} {t}^{2 k}\right) \frac{{e}^{-z t}}{{t}^{3}} \, 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) = \int_{0}^{\infty} \left(\frac{t}{2} \coth\!\left(\frac{t}{2}\right) - \sum_{k=0}^{N} \frac{B_{2 k}}{\left(2 k\right)!} {t}^{2 k}\right) \frac{{e}^{-z t}}{{t}^{3}} \, 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
Sumnf(n)\sum_{n} f(n) Sum
BernoulliBBnB_{n} Bernoulli number
Factorialn!n ! Factorial
Powab{a}^{b} Power
Expez{e}^{z} Exponential function
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), Integral(Mul(Sub(Mul(Div(t, 2), Coth(Div(t, 2))), Sum(Mul(Div(BernoulliB(Mul(2, k)), Factorial(Mul(2, k))), Pow(t, Mul(2, k))), For(k, 0, N))), Div(Exp(Neg(Mul(z, t))), Pow(t, 3))), For(t, 0, Infinity)))),
    Variables(z, N),
    Assumptions(And(Element(z, CC), Greater(Re(z), 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