Hypergeometric representations of Bessel functions - Fungrim: The Mathematical Functions Grimoire

ecd36f

J_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, -\frac{{z}^{2}}{4}\right)

Assumptions:

\nu \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

J_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, -\frac{{z}^{2}}{4}\right)

\nu \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, 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
`Hypergeometric0F1Regularized`	$\,{}_0{\textbf F}_1\!\left(a, z\right)$	Regularized confluent hypergeometric limit function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("ecd36f"),
    Formula(Equal(BesselJ(nu, z), Mul(Pow(Div(z, 2), nu), Hypergeometric0F1Regularized(Add(nu, 1), Neg(Div(Pow(z, 2), 4)))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, ZZGreaterEqual(0)), Element(z, CC)), And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

81eec6

I_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, \frac{{z}^{2}}{4}\right)

Assumptions:

\nu \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

I_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, \frac{{z}^{2}}{4}\right)

\nu \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, 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
`Hypergeometric0F1Regularized`	$\,{}_0{\textbf F}_1\!\left(a, z\right)$	Regularized confluent hypergeometric limit function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("81eec6"),
    Formula(Equal(BesselI(nu, z), Mul(Pow(Div(z, 2), nu), Hypergeometric0F1Regularized(Add(nu, 1), Div(Pow(z, 2), 4))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, ZZGreaterEqual(0)), Element(z, CC)), And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

b049dc

Y_{\nu}\!\left(z\right) = \frac{1}{\sin\!\left(\pi \nu\right)} \left(\cos\!\left(\pi \nu\right) {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, -\frac{{z}^{2}}{4}\right) - {\left(\frac{z}{2}\right)}^{-\nu} \,{}_0{\textbf F}_1\!\left(1 - \nu, -\frac{{z}^{2}}{4}\right)\right)

Assumptions:

\nu \in \mathbb{C} \setminus \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

Y_{\nu}\!\left(z\right) = \frac{1}{\sin\!\left(\pi \nu\right)} \left(\cos\!\left(\pi \nu\right) {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, -\frac{{z}^{2}}{4}\right) - {\left(\frac{z}{2}\right)}^{-\nu} \,{}_0{\textbf F}_1\!\left(1 - \nu, -\frac{{z}^{2}}{4}\right)\right)

\nu \in \mathbb{C} \setminus \mathbb{Z} \,\mathbin{\operatorname{and}}\, 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
`Sin`	$\sin\!\left(z\right)$	Sine
`ConstPi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`Hypergeometric0F1Regularized`	$\,{}_0{\textbf F}_1\!\left(a, z\right)$	Regularized confluent hypergeometric limit function
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("b049dc"),
    Formula(Equal(BesselY(nu, z), Mul(Div(1, Sin(Mul(ConstPi, nu))), Sub(Mul(Mul(Cos(Mul(ConstPi, nu)), Pow(Div(z, 2), nu)), Hypergeometric0F1Regularized(Add(nu, 1), Neg(Div(Pow(z, 2), 4)))), Mul(Pow(Div(z, 2), Neg(nu)), Hypergeometric0F1Regularized(Sub(1, nu), Neg(Div(Pow(z, 2), 4)))))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, SetMinus(CC, ZZ)), Element(z, SetMinus(CC, Set(0))))))

7efe21

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

Assumptions:

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

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

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, 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
`ConstPi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("7efe21"),
    Formula(Equal(BesselK(nu, z), Mul(Mul(Pow(Div(Mul(2, z), ConstPi), Neg(Div(1, 2))), Exp(Neg(z))), HypergeometricUStar(Add(nu, Div(1, 2)), Add(Mul(2, nu), 1), Mul(2, z))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

98703d

K_{\nu}\!\left(z\right) = \frac{1}{2} \frac{\pi}{\sin\!\left(\pi \nu\right)} \left({\left(\frac{z}{2}\right)}^{-\nu} \,{}_0{\textbf F}_1\!\left(1 - \nu, \frac{{z}^{2}}{4}\right) - {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(1 + \nu, \frac{{z}^{2}}{4}\right)\right)

Assumptions:

\nu \in \mathbb{C} \setminus \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

K_{\nu}\!\left(z\right) = \frac{1}{2} \frac{\pi}{\sin\!\left(\pi \nu\right)} \left({\left(\frac{z}{2}\right)}^{-\nu} \,{}_0{\textbf F}_1\!\left(1 - \nu, \frac{{z}^{2}}{4}\right) - {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(1 + \nu, \frac{{z}^{2}}{4}\right)\right)

\nu \in \mathbb{C} \setminus \mathbb{Z} \,\mathbin{\operatorname{and}}\, 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
`ConstPi`	$\pi$	The constant pi (3.14...)
`Sin`	$\sin\!\left(z\right)$	Sine
`Pow`	${a}^{b}$	Power
`Hypergeometric0F1Regularized`	$\,{}_0{\textbf F}_1\!\left(a, z\right)$	Regularized confluent hypergeometric limit function
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("98703d"),
    Formula(Equal(BesselK(nu, z), Mul(Mul(Div(1, 2), Div(ConstPi, Sin(Mul(ConstPi, nu)))), Sub(Mul(Pow(Div(z, 2), Neg(nu)), Hypergeometric0F1Regularized(Sub(1, nu), Div(Pow(z, 2), 4))), Mul(Pow(Div(z, 2), nu), Hypergeometric0F1Regularized(Add(1, nu), Div(Pow(z, 2), 4))))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, SetMinus(CC, ZZ)), Element(z, SetMinus(CC, Set(0))))))

9ad254

J_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \frac{{e}^{-i z}}{\Gamma\!\left(\nu + 1\right)} \,{}_1F_1\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)

Assumptions:

\nu \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \nu \notin \{-1, -2, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

J_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \frac{{e}^{-i z}}{\Gamma\!\left(\nu + 1\right)} \,{}_1F_1\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)

\nu \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \nu \notin \{-1, -2, \ldots\} \,\mathbin{\operatorname{and}}\, 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
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Hypergeometric1F1`	$\,{}_1F_1\!\left(a, b, z\right)$	Kummer confluent hypergeometric function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("9ad254"),
    Formula(Equal(BesselJ(nu, z), Mul(Mul(Pow(Div(z, 2), nu), Div(Exp(Neg(Mul(ConstI, z))), GammaFunction(Add(nu, 1)))), Hypergeometric1F1(Add(nu, Div(1, 2)), Add(Mul(2, nu), 1), Mul(Mul(2, ConstI), z))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, ZZGreaterEqual(0)), Element(z, CC)), And(Element(nu, CC), NotElement(nu, ZZLessEqual(-1)), Element(z, SetMinus(CC, Set(0))))))

32e162

J_{\nu}\!\left(z\right) = \frac{{z}^{\nu}}{{\left(2 \pi\right)}^{1 / 2}} \left({\left(i z\right)}^{-1 / 2 - \nu} {e}^{i z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, -2 i z\right) + {\left(-i z\right)}^{-1 / 2 - \nu} {e}^{-i z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)\right)

Assumptions:

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

J_{\nu}\!\left(z\right) = \frac{{z}^{\nu}}{{\left(2 \pi\right)}^{1 / 2}} \left({\left(i z\right)}^{-1 / 2 - \nu} {e}^{i z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, -2 i z\right) + {\left(-i z\right)}^{-1 / 2 - \nu} {e}^{-i z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)\right)

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, 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
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("32e162"),
    Formula(Equal(BesselJ(nu, z), Mul(Div(Pow(z, nu), Pow(Mul(2, ConstPi), Div(1, 2))), Add(Mul(Pow(Mul(ConstI, z), Sub(Neg(Div(1, 2)), nu)), Mul(Exp(Mul(ConstI, z)), HypergeometricUStar(Add(nu, Div(1, 2)), Add(Mul(2, nu), 1), Neg(Mul(Mul(2, ConstI), z))))), Mul(Pow(Neg(Mul(ConstI, z)), Sub(Neg(Div(1, 2)), nu)), Mul(Exp(Neg(Mul(ConstI, z))), HypergeometricUStar(Add(nu, Div(1, 2)), Add(Mul(2, nu), 1), Mul(Mul(2, ConstI), z)))))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

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}\!\left(z\right) \gt 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) \gt 0

Definitions:

Fungrim symbol	Notation	Short description
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`Pow`	${a}^{b}$	Power
`ConstPi`	$\pi$	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`	$\operatorname{Re}\!\left(z\right)$	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))))