# Fungrim entry: 80f7dc

$\left|\Gamma(z)\right| \le {\left(2 \pi\right)}^{1 / 2} \left|{z}^{z - 1 / 2} {e}^{-z}\right| \exp\!\left(\frac{1}{6 \left|z\right|}\right)$
Assumptions:$z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(z) \ge 0 \,\mathbin{\operatorname{and}}\, z \ne 0$
References:
• R. B. Paris and D. Kaminski (2001), Asymptotics of Mellin-Barnes integrals, Cambridge University Press. (2.1.18), p. 34.
TeX:
\left|\Gamma(z)\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}(z) \ge 0 \,\mathbin{\operatorname{and}}\, z \ne 0
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
Gamma$\Gamma(z)$ Gamma function
Pow${a}^{b}$ Power
Pi$\pi$ The constant pi (3.14...)
Exp${e}^{z}$ Exponential function
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("80f7dc"),
Formula(LessEqual(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(Div(1, Mul(6, Abs(z))))))),
Variables(z),
Assumptions(And(Element(z, CC), GreaterEqual(Re(z), 0), NotEqual(z, 0))),
References("R. B. Paris and D. Kaminski (2001), Asymptotics of Mellin-Barnes integrals, Cambridge University Press. (2.1.18), p. 34."))

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2020-01-31 18:09:28.494564 UTC