# Fungrim entry: d35c54

$\lim_{\varepsilon \to {0}^{+}} \left[\log G\!\left(x - \varepsilon i\right)\right] = \overline{\log G(x)} = \log G(x) - n \left(n - 1\right) \pi i\; \text{ where } n = \left\lfloor x \right\rfloor$
Assumptions:$x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x < 0 \;\mathbin{\operatorname{and}}\; x \notin \mathbb{Z}$
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
RightLimit$\lim_{x \to {a}^{+}} f(x)$ Limiting value, from the right
LogBarnesG$\log G(z)$ Logarithmic Barnes G-function
ConstI$i$ Imaginary unit
Conjugate$\overline{z}$ Complex conjugate
Pi$\pi$ The constant pi (3.14...)
RR$\mathbb{R}$ Real numbers
ZZ$\mathbb{Z}$ 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.

2021-03-15 19:12:00.328586 UTC