Fungrim entry: 5a11eb

\log G(x) = \begin{cases} \log\!\left(G(x)\right), & x > 0\\\log\!\left(\left|G(x)\right|\right) + \frac{1}{2} n \left(n - 1\right) \pi i, & \text{otherwise}\\ \end{cases}\; \text{ where } n = \left\lfloor x \right\rfloor

Assumptions:

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \notin \{0, -1, \ldots\}

TeX:

\log G(x) = \begin{cases} \log\!\left(G(x)\right), & x > 0\\\log\!\left(\left|G(x)\right|\right) + \frac{1}{2} n \left(n - 1\right) \pi i, & \text{otherwise}\\ \end{cases}\; \text{ where } n = \left\lfloor x \right\rfloor

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Log`	$\log(z)$	Natural logarithm
`BarnesG`	$G(z)$	Barnes G-function
`Abs`	$\left\|z\right\|$	Absolute value
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("5a11eb"),
    Formula(Equal(LogBarnesG(x), Where(Cases(Tuple(Log(BarnesG(x)), Greater(x, 0)), Tuple(Add(Log(Abs(BarnesG(x))), Mul(Mul(Mul(Mul(Div(1, 2), n), Sub(n, 1)), Pi), ConstI)), Otherwise)), Equal(n, Floor(x))))),
    Variables(x),
    Assumptions(And(Element(x, RR), NotElement(x, ZZLessEqual(0)))))

Topics using this entry

Barnes G-function