# 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)))))

## 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