Fungrim home page

Fungrim entry: 127f05

Jν ⁣(z)=(2πz)1/2(eiθU ⁣(ν+12,2ν+1,2iz)+eiθU ⁣(ν+12,2ν+1,2iz))   where θ=π(2ν+1)4zJ_{\nu}\!\left(z\right) = {\left(2 \pi z\right)}^{-1 / 2} \left({e}^{-i \theta} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, -2 i z\right) + {e}^{i \theta} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)\right)\; \text{ where } \theta = \frac{\pi \left(2 \nu + 1\right)}{4} - z
Assumptions:νCandzCandRe ⁣(z)>0\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(z\right) > 0
TeX:
J_{\nu}\!\left(z\right) = {\left(2 \pi z\right)}^{-1 / 2} \left({e}^{-i \theta} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, -2 i z\right) + {e}^{i \theta} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)\right)\; \text{ where } \theta = \frac{\pi \left(2 \nu + 1\right)}{4} - z

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(z\right) > 0
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...)
Expez{e}^{z} Exponential function
ConstIii Imaginary unit
HypergeometricUStarU ⁣(a,b,z)U^{*}\!\left(a, b, z\right) Scaled Tricomi confluent hypergeometric function
CCC\mathbb{C} Complex numbers
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
Source code for this entry:
Entry(ID("127f05"),
    Formula(Where(Equal(BesselJ(nu, z), Mul(Pow(Mul(Mul(2, ConstPi), z), Neg(Div(1, 2))), Add(Mul(Exp(Neg(Mul(ConstI, theta))), HypergeometricUStar(Add(nu, Div(1, 2)), Add(Mul(2, nu), 1), Neg(Mul(Mul(2, ConstI), z)))), Mul(Exp(Mul(ConstI, theta)), HypergeometricUStar(Add(nu, Div(1, 2)), Add(Mul(2, nu), 1), Mul(Mul(2, ConstI), z)))))), Equal(theta, Sub(Div(Mul(ConstPi, Add(Mul(2, nu), 1)), 4), z)))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, CC), Greater(Re(z), 0))))

Topics using this entry

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

2019-09-19 20:12:49.583742 UTC