Fungrim entry: 325a0e

\,{}_0{\textbf F}_1\!\left(a, z\right) = {\left(-z\right)}^{\left( 1 - a \right) / 2} J_{a - 1}\!\left(2 \sqrt{-z}\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \ne 0

TeX:

\,{}_0{\textbf F}_1\!\left(a, z\right) = {\left(-z\right)}^{\left( 1 - a \right) / 2} J_{a - 1}\!\left(2 \sqrt{-z}\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric0F1Regularized`	$\,{}_0{\textbf F}_1\!\left(a, z\right)$	Regularized confluent hypergeometric limit function
`Pow`	${a}^{b}$	Power
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("325a0e"),
    Formula(Equal(Hypergeometric0F1Regularized(a, z), Mul(Pow(Neg(z), Div(Sub(1, a), 2)), BesselJ(Sub(a, 1), Mul(2, Sqrt(Neg(z))))))),
    Variables(a, z),
    Assumptions(And(Element(a, CC), Element(z, CC), NotEqual(z, 0))))

Topics using this entry

Confluent hypergeometric functions