Fungrim home page

Fungrim entry: b049dc

Yν ⁣(z)=1sin ⁣(πν)(cos ⁣(πν)(z2)ν0F1 ⁣(ν+1,z24)(z2)ν0F1 ⁣(1ν,z24))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:νCZandzC{0}\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
BesselYYν ⁣(z)Y_{\nu}\!\left(z\right) Bessel function of the second kind
Sinsin(z)\sin(z) Sine
ConstPiπ\pi The constant pi (3.14...)
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("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

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

2019-10-05 13:11:19.856591 UTC