$\log G\!\left(1 + z\right) = \frac{\log\!\left(2 \pi\right) - 1}{2} z - \frac{1 + \gamma}{2} {z}^{2} + \sum_{n=3}^{\infty} \frac{{\left(-1\right)}^{n + 1} \zeta\!\left(n - 1\right)}{n} {z}^{n}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1$
TeX:
\log G\!\left(1 + z\right) = \frac{\log\!\left(2 \pi\right) - 1}{2} z - \frac{1 + \gamma}{2} {z}^{2} + \sum_{n=3}^{\infty} \frac{{\left(-1\right)}^{n + 1} \zeta\!\left(n - 1\right)}{n} {z}^{n}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1
Definitions:
Fungrim symbol Notation Short description
LogBarnesG$\log G(z)$ Logarithmic Barnes G-function
Log$\log(z)$ Natural logarithm
Pi$\pi$ The constant pi (3.14...)
ConstGamma$\gamma$ The constant gamma (0.577...)
Pow${a}^{b}$ Power
Sum$\sum_{n} f(n)$ Sum
RiemannZeta$\zeta\!\left(s\right)$ Riemann zeta function
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
Abs$\left|z\right|$ Absolute value
Source code for this entry:
Entry(ID("0ad263"),
Formula(Equal(LogBarnesG(Add(1, z)), Add(Sub(Mul(Div(Sub(Log(Mul(2, Pi)), 1), 2), z), Mul(Div(Add(1, ConstGamma), 2), Pow(z, 2))), Sum(Mul(Div(Mul(Pow(-1, Add(n, 1)), RiemannZeta(Sub(n, 1))), n), Pow(z, n)), For(n, 3, Infinity))))),
Variables(z),
Assumptions(And(Element(z, CC), Less(Abs(z), 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