Fungrim entry: 53026a

\Gamma(z) = \sqrt{2 \pi} \exp\!\left(\zeta'\!\left(0, z\right)\right)

Assumptions:

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

TeX:

\Gamma(z) = \sqrt{2 \pi} \exp\!\left(\zeta'\!\left(0, z\right)\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`Gamma`	$\Gamma(z)$	Gamma function
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("53026a"),
    Formula(Equal(Gamma(z), Mul(Sqrt(Mul(2, Pi)), Exp(HurwitzZeta(0, z, 1))))),
    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