Fungrim entry: 661054

\log \Gamma\!\left(1 + z\right) = -\gamma z + \sum_{k=2}^{\infty} \frac{\zeta\!\left(k\right)}{k} {\left(-z\right)}^{k}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|z\right| \lt 1

TeX:

\log \Gamma\!\left(1 + z\right) = -\gamma z + \sum_{k=2}^{\infty} \frac{\zeta\!\left(k\right)}{k} {\left(-z\right)}^{k}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|z\right| \lt 1

Definitions:

Fungrim symbol	Notation	Short description
`LogGamma`	$\log \Gamma\!\left(z\right)$	Logarithmic gamma function
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("661054"),
    Formula(Equal(LogGamma(Add(1, z)), Add(Neg(Mul(ConstGamma, z)), Sum(Mul(Div(RiemannZeta(k), k), Pow(Neg(z), k)), Tuple(k, 2, Infinity))))),
    Variables(z),
    Assumptions(And(Element(z, CC), Less(Abs(z), 1))))

Topics using this entry

Gamma function