# 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"),