Fungrim entry: 9ad254

J_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \frac{{e}^{-i z}}{\Gamma\!\left(\nu + 1\right)} \,{}_1F_1\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)

Assumptions:

\nu \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \nu \notin \{-1, -2, \ldots\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

J_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \frac{{e}^{-i z}}{\Gamma\!\left(\nu + 1\right)} \,{}_1F_1\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)

\nu \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \nu \notin \{-1, -2, \ldots\} \;\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
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`Gamma`	$\Gamma(z)$	Gamma function
`Hypergeometric1F1`	$\,{}_1F_1\!\left(a, b, z\right)$	Kummer confluent hypergeometric function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("9ad254"),
    Formula(Equal(BesselJ(nu, z), Mul(Mul(Pow(Div(z, 2), nu), Div(Exp(Neg(Mul(ConstI, z))), Gamma(Add(nu, 1)))), Hypergeometric1F1(Add(nu, Div(1, 2)), Add(Mul(2, nu), 1), Mul(Mul(2, ConstI), z))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, ZZGreaterEqual(0)), Element(z, CC)), And(Element(nu, CC), NotElement(nu, ZZLessEqual(-1)), Element(z, SetMinus(CC, Set(0))))))

Topics using this entry

Hypergeometric representations of Bessel functions