# Fungrim entry: e0b322

$\left|\Gamma\!\left(x + y i\right)\right| \ge \frac{\Gamma(x)}{\sqrt{\cosh\!\left(\pi y\right)}}$
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 \frac{\Gamma(x)}{\sqrt{\cosh\!\left(\pi y\right)}}

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
Sqrt$\sqrt{z}$ Principal square root
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("e0b322"),
Formula(GreaterEqual(Abs(Gamma(Add(x, Mul(y, ConstI)))), Div(Gamma(x), Sqrt(Cosh(Mul(Pi, y)))))),
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."))

## Topics using this entry

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2021-03-15 19:12:00.328586 UTC