# Fungrim entry: 685892

$J_{-1 / 3}\!\left(z\right) = \frac{1}{2 \omega} \left(3 \operatorname{Ai}\!\left(-{\omega}^{2}\right) + \sqrt{3} \operatorname{Bi}\!\left(-{\omega}^{2}\right)\right)\; \text{ where } \omega = {\left(\frac{3 z}{2}\right)}^{1 / 3}$
Assumptions:$z \in \mathbb{C} \setminus \left\{0\right\}$
TeX:
J_{-1 / 3}\!\left(z\right) = \frac{1}{2 \omega} \left(3 \operatorname{Ai}\!\left(-{\omega}^{2}\right) + \sqrt{3} \operatorname{Bi}\!\left(-{\omega}^{2}\right)\right)\; \text{ where } \omega = {\left(\frac{3 z}{2}\right)}^{1 / 3}

z \in \mathbb{C} \setminus \left\{0\right\}
Definitions:
Fungrim symbol Notation Short description
BesselJ$J_{\nu}\!\left(z\right)$ Bessel function of the first kind
AiryAi$\operatorname{Ai}\!\left(z\right)$ Airy function of the first kind
Pow${a}^{b}$ Power
Sqrt$\sqrt{z}$ Principal square root
AiryBi$\operatorname{Bi}\!\left(z\right)$ Airy function of the second kind
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("685892"),
Formula(Equal(BesselJ(Neg(Div(1, 3)), z), Where(Mul(Div(1, Mul(2, omega)), Add(Mul(3, AiryAi(Neg(Pow(omega, 2)))), Mul(Sqrt(3), AiryBi(Neg(Pow(omega, 2)))))), Equal(omega, Pow(Div(Mul(3, z), 2), Div(1, 3)))))),
Variables(z),
Assumptions(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.

2020-01-31 18:09:28.494564 UTC