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Fungrim entry: 7af1b9

Γ ⁣(x+yi)Γ ⁣(x)eπy/2\left|\Gamma\!\left(x + y i\right)\right| \ge \Gamma\!\left(x\right) {e}^{-\pi \left|y\right| / 2}
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 \Gamma\!\left(x\right) {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
Absz\left|z\right| Absolute value
GammaFunctionΓ ⁣(z)\Gamma\!\left(z\right) Gamma function
ConstIii Imaginary unit
Expez{e}^{z} Exponential function
ConstPiπ\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("7af1b9"),
    Formula(GreaterEqual(Abs(GammaFunction(Add(x, Mul(y, ConstI)))), Mul(GammaFunction(x), Exp(Neg(Div(Mul(ConstPi, 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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2019-09-19 20:12:49.583742 UTC