Fungrim entry: e72e96

J_{2 / 3}\!\left(z\right) = \frac{1}{2 {\omega}^{2}} \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_{2 / 3}\!\left(z\right) = \frac{1}{2 {\omega}^{2}} \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
`Pow`	${a}^{b}$	Power
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`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("e72e96"),
    Formula(Equal(BesselJ(Div(2, 3), z), Where(Mul(Div(1, Mul(2, Pow(omega, 2))), Add(Mul(3, AiryAi(Neg(Pow(omega, 2)), 1)), Mul(Sqrt(3), AiryBi(Neg(Pow(omega, 2)), 1)))), Equal(omega, Pow(Div(Mul(3, z), 2), Div(1, 3)))))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

Topics using this entry

Specific values of Bessel functions