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Fungrim entry: e0b322

Γ ⁣(x+yi)Γ(x)cosh ⁣(πy)\left|\Gamma\!\left(x + y i\right)\right| \ge \frac{\Gamma(x)}{\sqrt{\cosh\!\left(\pi y\right)}}
Assumptions:x[12,)andyRx \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
Absz\left|z\right| Absolute value
GammaΓ(z)\Gamma(z) Gamma function
ConstIii Imaginary unit
Sqrtz\sqrt{z} Principal square root
Piπ\pi The constant pi (3.14...)
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
Infinity\infty Positive infinity
RRR\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."))

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2019-11-19 15:10:20.037976 UTC