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Fungrim entry: 32e162

Jν ⁣(z)=zν(2π)1/2((iz)1/2νeizU ⁣(ν+12,2ν+1,2iz)+(iz)1/2νeizU ⁣(ν+12,2ν+1,2iz))J_{\nu}\!\left(z\right) = \frac{{z}^{\nu}}{{\left(2 \pi\right)}^{1 / 2}} \left({\left(i z\right)}^{-1 / 2 - \nu} {e}^{i z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, -2 i z\right) + {\left(-i z\right)}^{-1 / 2 - \nu} {e}^{-i z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)\right)
Assumptions:νCandzC{0}\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}
TeX:
J_{\nu}\!\left(z\right) = \frac{{z}^{\nu}}{{\left(2 \pi\right)}^{1 / 2}} \left({\left(i z\right)}^{-1 / 2 - \nu} {e}^{i z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, -2 i z\right) + {\left(-i z\right)}^{-1 / 2 - \nu} {e}^{-i z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)\right)

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}
Definitions:
Fungrim symbol Notation Short description
BesselJJν ⁣(z)J_{\nu}\!\left(z\right) Bessel function of the first kind
Powab{a}^{b} Power
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
Expez{e}^{z} Exponential function
HypergeometricUStarU ⁣(a,b,z)U^{*}\!\left(a, b, z\right) Scaled Tricomi confluent hypergeometric function
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("32e162"),
    Formula(Equal(BesselJ(nu, z), Mul(Div(Pow(z, nu), Pow(Mul(2, ConstPi), Div(1, 2))), Add(Mul(Pow(Mul(ConstI, z), Sub(Neg(Div(1, 2)), nu)), Mul(Exp(Mul(ConstI, z)), HypergeometricUStar(Add(nu, Div(1, 2)), Add(Mul(2, nu), 1), Neg(Mul(Mul(2, ConstI), z))))), Mul(Pow(Neg(Mul(ConstI, z)), Sub(Neg(Div(1, 2)), nu)), Mul(Exp(Neg(Mul(ConstI, z))), HypergeometricUStar(Add(nu, Div(1, 2)), Add(Mul(2, nu), 1), Mul(Mul(2, ConstI), z)))))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

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

2019-09-19 20:12:49.583742 UTC