Fungrim home page

Fungrim entry: 0ad263

logG ⁣(1+z)=log ⁣(2π)12z1+γ2z2+n=3(1)n+1ζ ⁣(n1)nzn\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:zC  and  z<1z \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
LogBarnesGlogG(z)\log G(z) Logarithmic Barnes G-function
Loglog(z)\log(z) Natural logarithm
Piπ\pi The constant pi (3.14...)
ConstGammaγ\gamma The constant gamma (0.577...)
Powab{a}^{b} Power
Sumnf(n)\sum_{n} f(n) Sum
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
Absz\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

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