# 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC