Fungrim entry: a787eb

\Gamma(z) \Gamma\!\left(z + \frac{1}{2}\right) = {2}^{1 - 2 z} \sqrt{\pi} \Gamma\!\left(2 z\right)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; 2 z \notin \{0, -1, \ldots\}

TeX:

\Gamma(z) \Gamma\!\left(z + \frac{1}{2}\right) = {2}^{1 - 2 z} \sqrt{\pi} \Gamma\!\left(2 z\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; 2 z \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`Gamma`	$\Gamma(z)$	Gamma function
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`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("a787eb"),
    Formula(Equal(Mul(Gamma(z), Gamma(Add(z, Div(1, 2)))), Mul(Mul(Pow(2, Sub(1, Mul(2, z))), Sqrt(Pi)), Gamma(Mul(2, z))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(Mul(2, z), ZZLessEqual(0)))))

Topics using this entry

Gamma 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