Fungrim entry: d56914

J_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(J_{\nu - 1}\!\left(z\right) + J_{\nu + 1}\!\left(z\right)\right)

Assumptions:

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

Alternative assumptions:

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

TeX:

J_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(J_{\nu - 1}\!\left(z\right) + J_{\nu + 1}\!\left(z\right)\right)

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

\nu \in \mathbb{C} \setminus \left\{0\right\} \,\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
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("d56914"),
    Formula(Equal(BesselJ(nu, z), Mul(Div(z, Mul(2, nu)), Add(BesselJ(Sub(nu, 1), z), BesselJ(Add(nu, 1), z))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, SetMinus(ZZ, Set(0))), Element(z, CC)), And(Element(nu, SetMinus(CC, Set(0))), Element(z, SetMinus(CC, Set(0))))))

Fungrim entry: d56914

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