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Fungrim entry: 6f8e14

logG ⁣(z+1)=z24+zlogΓ ⁣(z+1)(z(z+1)2+112)log(z)log(A)+n=1N1B2n+22n(2n+1)(2n+2)z2n+RN ⁣(z)\log G\!\left(z + 1\right) = \frac{{z}^{2}}{4} + z \log \Gamma\!\left(z + 1\right) - \left(\frac{z \left(z + 1\right)}{2} + \frac{1}{12}\right) \log(z) - \log(A) + \sum_{n=1}^{N - 1} \frac{B_{2 n + 2}}{2 n \left(2 n + 1\right) \left(2 n + 2\right) {z}^{2 n}} + R_{N}\!\left(z\right)
Assumptions:zC  and  z(,0]  and  NZ1z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}
References:
  • https://dx.doi.org/10.1098/rspa.2014.0534
TeX:
\log G\!\left(z + 1\right) = \frac{{z}^{2}}{4} + z \log \Gamma\!\left(z + 1\right) - \left(\frac{z \left(z + 1\right)}{2} + \frac{1}{12}\right) \log(z) - \log(A) + \sum_{n=1}^{N - 1} \frac{B_{2 n + 2}}{2 n \left(2 n + 1\right) \left(2 n + 2\right) {z}^{2 n}} + R_{N}\!\left(z\right)

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
LogBarnesGlogG(z)\log G(z) Logarithmic Barnes G-function
Powab{a}^{b} Power
LogGammalogΓ(z)\log \Gamma(z) Logarithmic gamma function
Loglog(z)\log(z) Natural logarithm
Sumnf(n)\sum_{n} f(n) Sum
BernoulliBBnB_{n} Bernoulli number
LogBarnesGRemainderRN ⁣(z)R_{N}\!\left(z\right) Remainder term in asymptotic expansion of logarithmic Barnes G-function
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("6f8e14"),
    Formula(Equal(LogBarnesG(Add(z, 1)), Add(Add(Sub(Sub(Add(Div(Pow(z, 2), 4), Mul(z, LogGamma(Add(z, 1)))), Mul(Add(Div(Mul(z, Add(z, 1)), 2), Div(1, 12)), Log(z))), Log(ConstGlaisher)), Sum(Div(BernoulliB(Add(Mul(2, n), 2)), Mul(Mul(Mul(Mul(2, n), Add(Mul(2, n), 1)), Add(Mul(2, n), 2)), Pow(z, Mul(2, n)))), For(n, 1, Sub(N, 1)))), LogBarnesGRemainder(N, z)))),
    Variables(z, N),
    Assumptions(And(Element(z, CC), NotElement(z, OpenClosedInterval(Neg(Infinity), 0)), Element(N, ZZGreaterEqual(1)))),
    References("https://dx.doi.org/10.1098/rspa.2014.0534"))

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

2020-08-27 09:56:25.682319 UTC