# Fungrim entry: 7af1b9

$\left|\Gamma\!\left(x + y i\right)\right| \ge \Gamma(x) {e}^{-\pi \left|y\right| / 2}$
Assumptions:$x \in \left[\frac{1}{2}, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{R}$
References:
• B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Inequality 3.10-4.
TeX:
\left|\Gamma\!\left(x + y i\right)\right| \ge \Gamma(x) {e}^{-\pi \left|y\right| / 2}

x \in \left[\frac{1}{2}, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{R}
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
Gamma$\Gamma(z)$ Gamma function
ConstI$i$ Imaginary unit
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
ClosedOpenInterval$\left[a, b\right)$ Closed-open interval
Infinity$\infty$ Positive infinity
RR$\mathbb{R}$ Real numbers
Source code for this entry:
Entry(ID("7af1b9"),
Formula(GreaterEqual(Abs(Gamma(Add(x, Mul(y, ConstI)))), Mul(Gamma(x), Exp(Neg(Div(Mul(Pi, Abs(y)), 2)))))),
Variables(x, y),
Assumptions(And(Element(x, ClosedOpenInterval(Div(1, 2), Infinity)), Element(y, RR))),
References("B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Inequality 3.10-4."))

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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