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Fungrim entry: 931d89

Γ(z)(2π)1/2zz1/2ezexp ⁣(16z)\left|\Gamma(z)\right| \ge {\left(2 \pi\right)}^{1 / 2} \left|{z}^{z - 1 / 2} {e}^{-z}\right| \exp\!\left(-\frac{1}{6 \left|z\right|}\right)
Assumptions:zCandRe(z)0andz0z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(z) \ge 0 \,\mathbin{\operatorname{and}}\, z \ne 0
TeX:
\left|\Gamma(z)\right| \ge {\left(2 \pi\right)}^{1 / 2} \left|{z}^{z - 1 / 2} {e}^{-z}\right| \exp\!\left(-\frac{1}{6 \left|z\right|}\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(z) \ge 0 \,\mathbin{\operatorname{and}}\, z \ne 0
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
GammaΓ(z)\Gamma(z) Gamma function
Powab{a}^{b} Power
Piπ\pi The constant pi (3.14...)
Expez{e}^{z} Exponential function
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
Entry(ID("931d89"),
    Formula(GreaterEqual(Abs(Gamma(z)), Mul(Mul(Pow(Mul(2, Pi), Div(1, 2)), Abs(Mul(Pow(z, Sub(z, Div(1, 2))), Exp(Neg(z))))), Exp(Neg(Div(1, Mul(6, Abs(z)))))))),
    Variables(z),
    Assumptions(And(Element(z, CC), GreaterEqual(Re(z), 0), NotEqual(z, 0))))

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

2020-01-31 18:09:28.494564 UTC