Fungrim entry: 32e162

J_{\nu}\!\left(z\right) = \frac{{z}^{\nu}}{{\left(2 \pi\right)}^{1 / 2}} \left({\left(i z\right)}^{-1 / 2 - \nu} {e}^{i z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, -2 i z\right) + {\left(-i z\right)}^{-1 / 2 - \nu} {e}^{-i z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)\right)

Assumptions:

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

J_{\nu}\!\left(z\right) = \frac{{z}^{\nu}}{{\left(2 \pi\right)}^{1 / 2}} \left({\left(i z\right)}^{-1 / 2 - \nu} {e}^{i z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, -2 i z\right) + {\left(-i z\right)}^{-1 / 2 - \nu} {e}^{-i z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)\right)

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`Pow`	${a}^{b}$	Power
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("32e162"),
    Formula(Equal(BesselJ(nu, z), Mul(Div(Pow(z, nu), Pow(Mul(2, ConstPi), Div(1, 2))), Add(Mul(Pow(Mul(ConstI, z), Sub(Neg(Div(1, 2)), nu)), Mul(Exp(Mul(ConstI, z)), HypergeometricUStar(Add(nu, Div(1, 2)), Add(Mul(2, nu), 1), Neg(Mul(Mul(2, ConstI), z))))), Mul(Pow(Neg(Mul(ConstI, z)), Sub(Neg(Div(1, 2)), nu)), Mul(Exp(Neg(Mul(ConstI, z))), HypergeometricUStar(Add(nu, Div(1, 2)), Add(Mul(2, nu), 1), Mul(Mul(2, ConstI), z)))))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

Topics using this entry

Hypergeometric representations of Bessel functions