Fungrim home page

Fungrim entry: d35c54

limε0+[logG ⁣(xεi)]=logG(x)=logG(x)n(n1)πi   where n=x\lim_{\varepsilon \to {0}^{+}} \left[\log G\!\left(x - \varepsilon i\right)\right] = \overline{\log G(x)} = \log G(x) - n \left(n - 1\right) \pi i\; \text{ where } n = \left\lfloor x \right\rfloor
Assumptions:xRandx<0andxZx \in \mathbb{R} \,\mathbin{\operatorname{and}}\, x < 0 \,\mathbin{\operatorname{and}}\, x \notin \mathbb{Z}
TeX:
\lim_{\varepsilon \to {0}^{+}} \left[\log G\!\left(x - \varepsilon i\right)\right] = \overline{\log G(x)} = \log G(x) - n \left(n - 1\right) \pi i\; \text{ where } n = \left\lfloor x \right\rfloor

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, x < 0 \,\mathbin{\operatorname{and}}\, x \notin \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
RightLimitlimxa+f(x)\lim_{x \to {a}^{+}} f(x) Limiting value, from the right
LogBarnesGlogG(z)\log G(z) Logarithmic Barnes G-function
ConstIii Imaginary unit
Conjugatez\overline{z} Complex conjugate
Piπ\pi The constant pi (3.14...)
RRR\mathbb{R} Real numbers
ZZZ\mathbb{Z} Integers
Source code for this entry:
Entry(ID("d35c54"),
    Formula(Equal(RightLimit(Brackets(LogBarnesG(Sub(x, Mul(epsilon, ConstI)))), For(epsilon, 0)), Conjugate(LogBarnesG(x)), Where(Sub(LogBarnesG(x), Mul(Mul(Mul(n, Sub(n, 1)), Pi), ConstI)), Equal(n, Floor(x))))),
    Variables(x),
    Assumptions(And(Element(x, RR), Less(x, 0), NotElement(x, ZZ))))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-01-31 18:09:28.494564 UTC