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Fungrim entry: cc4572

Jν ⁣(z)1Γ ⁣(ν+1)z2νexp ⁣(Im ⁣(z))\left|J_{\nu}\!\left(z\right)\right| \le \frac{1}{\Gamma\!\left(\nu + 1\right)} {\left|\frac{z}{2}\right|}^{\nu} \exp\!\left(\left|\operatorname{Im}\!\left(z\right)\right|\right)
Assumptions:ν[12,)andzC{0}\nu \in \left[-\frac{1}{2}, \infty\right) \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}
TeX:
\left|J_{\nu}\!\left(z\right)\right| \le \frac{1}{\Gamma\!\left(\nu + 1\right)} {\left|\frac{z}{2}\right|}^{\nu} \exp\!\left(\left|\operatorname{Im}\!\left(z\right)\right|\right)

\nu \in \left[-\frac{1}{2}, \infty\right) \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
BesselJJν ⁣(z)J_{\nu}\!\left(z\right) Bessel function of the first kind
GammaFunctionΓ ⁣(z)\Gamma\!\left(z\right) Gamma function
Powab{a}^{b} Power
Expez{e}^{z} Exponential function
ImIm ⁣(z)\operatorname{Im}\!\left(z\right) Imaginary part
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("cc4572"),
    Formula(LessEqual(Abs(BesselJ(nu, z)), Mul(Mul(Div(1, GammaFunction(Add(nu, 1))), Pow(Abs(Div(z, 2)), nu)), Exp(Abs(Im(z)))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, ClosedOpenInterval(Neg(Div(1, 2)), Infinity)), Element(z, SetMinus(CC, Set(0))))))

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2019-09-19 20:12:49.583742 UTC