Specific values of Bessel functions

621a9b

J_{-1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\cos(z)}{z}

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

J_{-1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\cos(z)}{z}

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
`Pi`	$\pi$	The constant pi (3.14...)
`Cos`	$\cos(z)$	Cosine
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("621a9b"),
    Formula(Equal(BesselJ(Neg(Div(1, 2)), z), Mul(Pow(Div(Mul(2, z), Pi), Div(1, 2)), Div(Cos(z), z)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

121b21

J_{1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\sin(z)}{z}

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

J_{1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\sin(z)}{z}

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
`Pi`	$\pi$	The constant pi (3.14...)
`Sin`	$\sin(z)$	Sine
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("121b21"),
    Formula(Equal(BesselJ(Div(1, 2), z), Mul(Pow(Div(Mul(2, z), Pi), Div(1, 2)), Div(Sin(z), z)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

a2a294

J_{3 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \left(\frac{\sin(z)}{{z}^{2}} - \frac{\cos(z)}{z}\right)

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

J_{3 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \left(\frac{\sin(z)}{{z}^{2}} - \frac{\cos(z)}{z}\right)

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
`Pi`	$\pi$	The constant pi (3.14...)
`Sin`	$\sin(z)$	Sine
`Cos`	$\cos(z)$	Cosine
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("a2a294"),
    Formula(Equal(BesselJ(Div(3, 2), z), Mul(Pow(Div(Mul(2, z), Pi), Div(1, 2)), Sub(Div(Sin(z), Pow(z, 2)), Div(Cos(z), z))))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

5679f2

Y_{-1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\sin(z)}{z}

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

Y_{-1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\sin(z)}{z}

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselY`	$Y_{\nu}\!\left(z\right)$	Bessel function of the second kind
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Sin`	$\sin(z)$	Sine
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("5679f2"),
    Formula(Equal(BesselY(Neg(Div(1, 2)), z), Mul(Pow(Div(Mul(2, z), Pi), Div(1, 2)), Div(Sin(z), z)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

4dfd41

Y_{1 / 2}\!\left(z\right) = -{\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\cos(z)}{z}

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

Y_{1 / 2}\!\left(z\right) = -{\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\cos(z)}{z}

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselY`	$Y_{\nu}\!\left(z\right)$	Bessel function of the second kind
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Cos`	$\cos(z)$	Cosine
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4dfd41"),
    Formula(Equal(BesselY(Div(1, 2), z), Neg(Mul(Pow(Div(Mul(2, z), Pi), Div(1, 2)), Div(Cos(z), z))))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

8472cc

Y_{3 / 2}\!\left(z\right) = -{\left(\frac{2 z}{\pi}\right)}^{1 / 2} \left(\frac{\cos(z)}{{z}^{2}} + \frac{\sin(z)}{z}\right)

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

Y_{3 / 2}\!\left(z\right) = -{\left(\frac{2 z}{\pi}\right)}^{1 / 2} \left(\frac{\cos(z)}{{z}^{2}} + \frac{\sin(z)}{z}\right)

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselY`	$Y_{\nu}\!\left(z\right)$	Bessel function of the second kind
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Cos`	$\cos(z)$	Cosine
`Sin`	$\sin(z)$	Sine
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("8472cc"),
    Formula(Equal(BesselY(Div(3, 2), z), Neg(Mul(Pow(Div(Mul(2, z), Pi), Div(1, 2)), Add(Div(Cos(z), Pow(z, 2)), Div(Sin(z), z)))))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

5d9c43

I_{-1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\cosh(z)}{z}

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

I_{-1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\cosh(z)}{z}

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("5d9c43"),
    Formula(Equal(BesselI(Neg(Div(1, 2)), z), Mul(Pow(Div(Mul(2, z), Pi), Div(1, 2)), Div(Cosh(z), z)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

a59981

I_{1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\sinh(z)}{z}

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

I_{1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\sinh(z)}{z}

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("a59981"),
    Formula(Equal(BesselI(Div(1, 2), z), Mul(Pow(Div(Mul(2, z), Pi), Div(1, 2)), Div(Sinh(z), z)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

65647f

I_{3 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \left(\frac{\cosh(z)}{z} - \frac{\sinh(z)}{{z}^{2}}\right)

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

I_{3 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \left(\frac{\cosh(z)}{z} - \frac{\sinh(z)}{{z}^{2}}\right)

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("65647f"),
    Formula(Equal(BesselI(Div(3, 2), z), Mul(Pow(Div(Mul(2, z), Pi), Div(1, 2)), Sub(Div(Cosh(z), z), Div(Sinh(z), Pow(z, 2)))))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

7ac286

K_{-1 / 2}\!\left(z\right) = {\left(\frac{\pi z}{2}\right)}^{1 / 2} \frac{{e}^{-z}}{z}

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

K_{-1 / 2}\!\left(z\right) = {\left(\frac{\pi z}{2}\right)}^{1 / 2} \frac{{e}^{-z}}{z}

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("7ac286"),
    Formula(Equal(BesselK(Neg(Div(1, 2)), z), Mul(Pow(Div(Mul(Pi, z), 2), Div(1, 2)), Div(Exp(Neg(z)), z)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

d1f5c5

K_{1 / 2}\!\left(z\right) = {\left(\frac{\pi z}{2}\right)}^{1 / 2} \frac{{e}^{-z}}{z}

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

K_{1 / 2}\!\left(z\right) = {\left(\frac{\pi z}{2}\right)}^{1 / 2} \frac{{e}^{-z}}{z}

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("d1f5c5"),
    Formula(Equal(BesselK(Div(1, 2), z), Mul(Pow(Div(Mul(Pi, z), 2), Div(1, 2)), Div(Exp(Neg(z)), z)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

0c09cc

K_{3 / 2}\!\left(z\right) = {\left(\frac{\pi z}{2}\right)}^{1 / 2} {e}^{-z} \left(\frac{1}{z} + \frac{1}{{z}^{2}}\right)

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

K_{3 / 2}\!\left(z\right) = {\left(\frac{\pi z}{2}\right)}^{1 / 2} {e}^{-z} \left(\frac{1}{z} + \frac{1}{{z}^{2}}\right)

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("0c09cc"),
    Formula(Equal(BesselK(Div(3, 2), z), Mul(Pow(Div(Mul(Pi, z), 2), Div(1, 2)), Mul(Exp(Neg(z)), Add(Div(1, z), Div(1, Pow(z, 2))))))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

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)))))

d39c46

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("d39c46"),
    Formula(Equal(BesselJ(Div(1, 3), z), Where(Mul(Div(1, Mul(2, omega)), Sub(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)))))

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)))))

fda595

K_{-1 / 3}\!\left(z\right) = \frac{\sqrt{3} \pi}{\omega} \operatorname{Ai}\!\left({\omega}^{2}\right)\; \text{ where } \omega = {\left(\frac{3 z}{2}\right)}^{1 / 3}

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

K_{-1 / 3}\!\left(z\right) = \frac{\sqrt{3} \pi}{\omega} \operatorname{Ai}\!\left({\omega}^{2}\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
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("fda595"),
    Formula(Equal(BesselK(Neg(Div(1, 3)), z), Where(Mul(Div(Mul(Sqrt(3), Pi), omega), AiryAi(Pow(omega, 2))), Equal(omega, Pow(Div(Mul(3, z), 2), Div(1, 3)))))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

49d754

K_{1 / 3}\!\left(z\right) = \frac{\sqrt{3} \pi}{\omega} \operatorname{Ai}\!\left({\omega}^{2}\right)\; \text{ where } \omega = {\left(\frac{3 z}{2}\right)}^{1 / 3}

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

K_{1 / 3}\!\left(z\right) = \frac{\sqrt{3} \pi}{\omega} \operatorname{Ai}\!\left({\omega}^{2}\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
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("49d754"),
    Formula(Equal(BesselK(Div(1, 3), z), Where(Mul(Div(Mul(Sqrt(3), Pi), omega), AiryAi(Pow(omega, 2))), Equal(omega, Pow(Div(Mul(3, z), 2), Div(1, 3)))))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

c362e8

K_{2 / 3}\!\left(z\right) = -\frac{\sqrt{3} \pi}{{\omega}^{2}} \operatorname{Ai}'\!\left({\omega}^{2}\right)\; \text{ where } \omega = {\left(\frac{3 z}{2}\right)}^{1 / 3}

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

K_{2 / 3}\!\left(z\right) = -\frac{\sqrt{3} \pi}{{\omega}^{2}} \operatorname{Ai}'\!\left({\omega}^{2}\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
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("c362e8"),
    Formula(Equal(BesselK(Div(2, 3), z), Where(Neg(Mul(Div(Mul(Sqrt(3), Pi), Pow(omega, 2)), AiryAi(Pow(omega, 2), 1))), Equal(omega, Pow(Div(Mul(3, z), 2), Div(1, 3)))))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

Trigonometric cases

Hyperbolic cases

Airy function cases