Fungrim entry: cc4572

\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:

\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
`Abs`	$\left\|z\right\|$	Absolute value
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	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))))))

Topics using this entry

Bessel functions