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Fungrim entry: 98703d

Kν ⁣(z)=12πsin ⁣(πν)((z2)ν0F1 ⁣(1ν,z24)(z2)ν0F1 ⁣(1+ν,z24))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:νCZandzC{0}\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
BesselKKν ⁣(z)K_{\nu}\!\left(z\right) Modified Bessel function of the second kind
ConstPiπ\pi The constant pi (3.14...)
Sinsin ⁣(z)\sin\!\left(z\right) Sine
Powab{a}^{b} Power
Hypergeometric0F1Regularized0F1 ⁣(a,z)\,{}_0{\textbf F}_1\!\left(a, z\right) Regularized confluent hypergeometric limit function
CCC\mathbb{C} Complex numbers
ZZZ\mathbb{Z} Integers
Source code for this entry:
Entry(ID("98703d"),
    Formula(Equal(BesselK(nu, z), Mul(Mul(Div(1, 2), Div(ConstPi, Sin(Mul(ConstPi, 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))))))

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2019-08-25 15:30:03.056001 UTC