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Fungrim entry: 9ad254

Jν ⁣(z)=(z2)νeizΓ ⁣(ν+1)1F1 ⁣(ν+12,2ν+1,2iz)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:νZ0  and  zC\nu \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
Alternative assumptions:νC  and  ν{1,2,}  and  zC{0}\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \nu \notin \{-1, -2, \ldots\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}
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\}
Fungrim symbol Notation Short description
BesselJJν ⁣(z)J_{\nu}\!\left(z\right) Bessel function of the first kind
Powab{a}^{b} Power
Expez{e}^{z} Exponential function
ConstIii Imaginary unit
GammaΓ(z)\Gamma(z) Gamma function
Hypergeometric1F11F1 ⁣(a,b,z)\,{}_1F_1\!\left(a, b, z\right) Kummer confluent hypergeometric function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
    Formula(Equal(BesselJ(nu, z), Mul(Mul(Pow(Div(z, 2), nu), Div(Exp(Neg(Mul(ConstI, z))), Gamma(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))))))

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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