Fungrim home page

Fungrim entry: 80f7dc

Γ ⁣(z)(2π)1/2zz1/2ezexp ⁣(16z)\left|\Gamma\!\left(z\right)\right| \le {\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}\!\left(z\right) \ge 0 \,\mathbin{\operatorname{and}}\, z \ne 0
References:
  • R. B. Paris and D. Kaminski (2001), Asymptotics of Mellin-Barnes integrals, Cambridge University Press. (2.1.18), p. 34.
TeX:
\left|\Gamma\!\left(z\right)\right| \le {\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}\!\left(z\right) \ge 0 \,\mathbin{\operatorname{and}}\, z \ne 0
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
GammaFunctionΓ ⁣(z)\Gamma\!\left(z\right) Gamma function
Powab{a}^{b} Power
ConstPiπ\pi The constant pi (3.14...)
Expez{e}^{z} Exponential function
CCC\mathbb{C} Complex numbers
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
Source code for this entry:
Entry(ID("80f7dc"),
    Formula(LessEqual(Abs(GammaFunction(z)), Mul(Mul(Pow(Mul(2, ConstPi), Div(1, 2)), Abs(Mul(Pow(z, Sub(z, Div(1, 2))), Exp(Neg(z))))), Exp(Div(1, Mul(6, Abs(z))))))),
    Variables(z),
    Assumptions(And(Element(z, CC), GreaterEqual(Re(z), 0), Unequal(z, 0))),
    References("R. B. Paris and D. Kaminski (2001), Asymptotics of Mellin-Barnes integrals, Cambridge University Press. (2.1.18), p. 34."))

Topics using this entry

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

2019-09-22 15:43:45.410764 UTC