Fungrim entry: b049dc

Y_{\nu}\!\left(z\right) = \frac{1}{\sin\!\left(\pi \nu\right)} \left(\cos\!\left(\pi \nu\right) {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, -\frac{{z}^{2}}{4}\right) - {\left(\frac{z}{2}\right)}^{-\nu} \,{}_0{\textbf F}_1\!\left(1 - \nu, -\frac{{z}^{2}}{4}\right)\right)

Assumptions:

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

TeX:

Y_{\nu}\!\left(z\right) = \frac{1}{\sin\!\left(\pi \nu\right)} \left(\cos\!\left(\pi \nu\right) {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, -\frac{{z}^{2}}{4}\right) - {\left(\frac{z}{2}\right)}^{-\nu} \,{}_0{\textbf F}_1\!\left(1 - \nu, -\frac{{z}^{2}}{4}\right)\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`BesselY`	$Y_{\nu}\!\left(z\right)$	Bessel function of the second kind
`Sin`	$\sin\!\left(z\right)$	Sine
`ConstPi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`Hypergeometric0F1Regularized`	$\,{}_0{\textbf F}_1\!\left(a, z\right)$	Regularized confluent hypergeometric limit function
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("b049dc"),
    Formula(Equal(BesselY(nu, z), Mul(Div(1, Sin(Mul(ConstPi, nu))), Sub(Mul(Mul(Cos(Mul(ConstPi, nu)), Pow(Div(z, 2), nu)), Hypergeometric0F1Regularized(Add(nu, 1), Neg(Div(Pow(z, 2), 4)))), Mul(Pow(Div(z, 2), Neg(nu)), Hypergeometric0F1Regularized(Sub(1, nu), Neg(Div(Pow(z, 2), 4)))))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, SetMinus(CC, ZZ)), Element(z, SetMinus(CC, Set(0))))))

Topics using this entry

Hypergeometric representations of Bessel functions