Fungrim entry: 7efe21

K_{\nu}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{-1 / 2} {e}^{-z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 z\right)

Assumptions:

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

TeX:

K_{\nu}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{-1 / 2} {e}^{-z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 z\right)

\nu \in \mathbb{C} \,\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
`Pow`	${a}^{b}$	Power
`ConstPi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("7efe21"),
    Formula(Equal(BesselK(nu, z), Mul(Mul(Pow(Div(Mul(2, z), ConstPi), Neg(Div(1, 2))), Exp(Neg(z))), HypergeometricUStar(Add(nu, Div(1, 2)), Add(Mul(2, nu), 1), Mul(2, z))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

Fungrim entry: 7efe21

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