Fungrim entry: f3b870

\log \Gamma(z) = \zeta'\!\left(0, z\right) + \frac{\log\!\left(2 \pi\right)}{2}

Assumptions:

z \in \mathbb{C} \setminus \{0, -1, \ldots\}

TeX:

\log \Gamma(z) = \zeta'\!\left(0, z\right) + \frac{\log\!\left(2 \pi\right)}{2}

z \in \mathbb{C} \setminus \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("f3b870"),
    Formula(Equal(LogGamma(z), Add(HurwitzZeta(0, z, 1), Div(Log(Mul(2, Pi)), 2)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, ZZLessEqual(0)))))

Topics using this entry

Hurwitz zeta function

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