Fungrim home page

Fungrim entry: e05807

logG(z)=(z1)logΓ(z)ζ ⁣(1,z)+ζ(1)\log G(z) = \left(z - 1\right) \log \Gamma(z) - \zeta'\!\left(-1, z\right) + \zeta'(-1)
Assumptions:zC  and  z{0,1,}z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}
\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\}
Fungrim symbol Notation Short description
LogBarnesGlogG(z)\log G(z) Logarithmic Barnes G-function
LogGammalogΓ(z)\log \Gamma(z) Logarithmic gamma function
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
    Formula(Equal(LogBarnesG(z), Add(Sub(Mul(Sub(z, 1), LogGamma(z)), HurwitzZeta(-1, z, 1)), ComplexDerivative(RiemannZeta(s), For(s, -1))))),
    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