Fungrim entry: 98703d

K_{\nu}\!\left(z\right) = \frac{1}{2} \frac{\pi}{\sin\!\left(\pi \nu\right)} \left({\left(\frac{z}{2}\right)}^{-\nu} \,{}_0{\textbf F}_1\!\left(1 - \nu, \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:

K_{\nu}\!\left(z\right) = \frac{1}{2} \frac{\pi}{\sin\!\left(\pi \nu\right)} \left({\left(\frac{z}{2}\right)}^{-\nu} \,{}_0{\textbf F}_1\!\left(1 - \nu, \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
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`Pi`	$\pi$	The constant pi (3.14...)
`Sin`	$\sin(z)$	Sine
`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("98703d"),
    Formula(Equal(BesselK(nu, z), Mul(Mul(Div(1, 2), Div(Pi, Sin(Mul(Pi, nu)))), Sub(Mul(Pow(Div(z, 2), Neg(nu)), Hypergeometric0F1Regularized(Sub(1, nu), Div(Pow(z, 2), 4))), Mul(Pow(Div(z, 2), nu), Hypergeometric0F1Regularized(Add(1, nu), 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