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Fungrim entry: b7fec0

Γ(z)(2π)1/2zx1/2eπy/2exp ⁣(16z)   where z=x+yi\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[0,)  and  yR  and  x+yi0x \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
Absz\left|z\right| Absolute value
GammaΓ(z)\Gamma(z) Gamma function
Powab{a}^{b} Power
Piπ\pi The constant pi (3.14...)
Expez{e}^{z} Exponential function
ConstIii Imaginary unit
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("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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2020-04-08 16:14:44.404316 UTC