# Fungrim entry: 127f05

$J_{\nu}\!\left(z\right) = {\left(2 \pi z\right)}^{-1 / 2} \left({e}^{-i \theta} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, -2 i z\right) + {e}^{i \theta} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)\right)\; \text{ where } \theta = \frac{\pi \left(2 \nu + 1\right)}{4} - z$
Assumptions:$\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0$
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}(z) > 0
Definitions:
Fungrim symbol Notation Short description
BesselJ$J_{\nu}\!\left(z\right)$ Bessel function of the first kind
Pow${a}^{b}$ Power
Pi$\pi$ The constant pi (3.14...)
Exp${e}^{z}$ Exponential function
ConstI$i$ Imaginary unit
HypergeometricUStar$U^{*}\!\left(a, b, z\right)$ Scaled Tricomi confluent hypergeometric function
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("127f05"),
Formula(Where(Equal(BesselJ(nu, z), Mul(Pow(Mul(Mul(2, Pi), 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(Pi, 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.

2021-03-15 19:12:00.328586 UTC