Fungrim home page

Fungrim entry: 95f771

logG ⁣(z+1)=z(1z)2+z2log ⁣(2π)+0zxψ ⁣(x)dx\log G\!\left(z + 1\right) = \frac{z \left(1 - z\right)}{2} + \frac{z}{2} \log\!\left(2 \pi\right) + \int_{0}^{z} x \psi\!\left(x\right) \, dx
Assumptions:zC  and  z(,1]z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, -1\right]
References:
  • https://arxiv.org/abs/math/0308086
TeX:
\log G\!\left(z + 1\right) = \frac{z \left(1 - z\right)}{2} + \frac{z}{2} \log\!\left(2 \pi\right) + \int_{0}^{z} x \psi\!\left(x\right) \, dx

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, -1\right]
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...)
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
Source code for this entry:
Entry(ID("95f771"),
    Formula(Equal(LogBarnesG(Add(z, 1)), Add(Add(Div(Mul(z, Sub(1, z)), 2), Mul(Div(z, 2), Log(Mul(2, Pi)))), Integral(Mul(x, DigammaFunction(x)), For(x, 0, z))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, OpenClosedInterval(Neg(Infinity), -1)))),
    References("https://arxiv.org/abs/math/0308086"))

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