Fungrim entry: b7fec0

\left|\Gamma(z)\right| \le {\left(2 \pi\right)}^{1 / 2} {\left|z\right|}^{x - 1 / 2} {e}^{-\pi \left|y\right| / 2} \exp\!\left(\frac{1}{6 \left|z\right|}\right)\; \text{ where } z = x + y i

Assumptions:

x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x + y i \ne 0

References:

R. B. Paris and D. Kaminski (2001), Asymptotics of Mellin-Barnes integrals, Cambridge University Press. (2.1.19), p. 34.

TeX:

\left|\Gamma(z)\right| \le {\left(2 \pi\right)}^{1 / 2} {\left|z\right|}^{x - 1 / 2} {e}^{-\pi \left|y\right| / 2} \exp\!\left(\frac{1}{6 \left|z\right|}\right)\; \text{ where } z = x + y i

x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x + y i \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
`ConstI`	$i$	Imaginary unit
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("b7fec0"),
    Formula(Where(LessEqual(Abs(Gamma(z)), Mul(Mul(Mul(Pow(Mul(2, Pi), Div(1, 2)), Pow(Abs(z), Sub(x, Div(1, 2)))), Exp(Neg(Div(Mul(Pi, Abs(y)), 2)))), Exp(Div(1, Mul(6, Abs(z)))))), Equal(z, Add(x, Mul(y, ConstI))))),
    Variables(x, y),
    Assumptions(And(Element(x, ClosedOpenInterval(0, Infinity)), Element(y, RR), NotEqual(Add(x, Mul(y, ConstI)), 0))),
    References("R. B. Paris and D. Kaminski (2001), Asymptotics of Mellin-Barnes integrals, Cambridge University Press. (2.1.19), p. 34."))

Fungrim entry: b7fec0

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