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Fungrim entry: a787eb

Γ(z)Γ ⁣(z+12)=212zπΓ ⁣(2z)\Gamma(z) \Gamma\!\left(z + \frac{1}{2}\right) = {2}^{1 - 2 z} \sqrt{\pi} \Gamma\!\left(2 z\right)
Assumptions:zCand2z{0,1,}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
GammaFunctionΓ(z)\Gamma(z) Gamma function
Powab{a}^{b} Power
Sqrtz\sqrt{z} Principal square root
ConstPiπ\pi The constant pi (3.14...)
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
Entry(ID("a787eb"),
    Formula(Equal(Mul(GammaFunction(z), GammaFunction(Add(z, Div(1, 2)))), Mul(Mul(Pow(2, Sub(1, Mul(2, z))), Sqrt(ConstPi)), GammaFunction(Mul(2, z))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(Mul(2, z), ZZLessEqual(0)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC