# 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."))

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

2021-03-15 19:12:00.328586 UTC