Fungrim entry: 0ad263

\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

Barnes G-function