Fungrim entry: 6f8e14

\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:

z \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
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Pow`	${a}^{b}$	Power
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`Log`	$\log(z)$	Natural logarithm
`Sum`	$\sum_{n} f(n)$	Sum
`BernoulliB`	$B_{n}$	Bernoulli number
`LogBarnesGRemainder`	$R_{N}\!\left(z\right)$	Remainder term in asymptotic expansion of logarithmic Barnes G-function
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	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"))

Topics using this entry

Barnes G-function