Fungrim entry: 80f7dc

\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:

z \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
`Abs`	$\left\|z\right\|$	Absolute value
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Pow`	${a}^{b}$	Power
`ConstPi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}\!\left(z\right)$	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

Bounds and inequalities for the gamma function