Fungrim entry: e05807

\log G(z) = \left(z - 1\right) \log \Gamma(z) - \zeta'\!\left(-1, z\right) + \zeta'(-1)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

TeX:

\log G(z) = \left(z - 1\right) \log \Gamma(z) - \zeta'\!\left(-1, z\right) + \zeta'(-1)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("e05807"),
    Formula(Equal(LogBarnesG(z), Add(Sub(Mul(Sub(z, 1), LogGamma(z)), HurwitzZeta(-1, z, 1)), ComplexDerivative(RiemannZeta(s), For(s, -1))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, ZZLessEqual(0)))))

Fungrim entry: e05807

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