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Fungrim entry: d280c5

F,η ⁣(z)=C ⁣(η)z+1eωiz1F1 ⁣(1++ωiη,2+2,2ωiz)F_{\ell,\eta}\!\left(z\right) = C_{\ell}\!\left(\eta\right) {z}^{\ell + 1} {e}^{\omega i z} \,{}_1F_1\!\left(1 + \ell + \omega i \eta, 2 \ell + 2, -2 \omega i z\right)
Assumptions:ω{1,1}andCandηCand(1++iη{0,1,}and1+iη{0,1,})and2+2{0,1,}andzC{0}\omega \in \left\{-1, 1\right\} \,\mathbin{\operatorname{and}}\, \ell \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \eta \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(1 + \ell + i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, 1 + \ell - i \eta \notin \{0, -1, \ldots\}\right) \,\mathbin{\operatorname{and}}\, 2 \ell + 2 \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}
TeX:
F_{\ell,\eta}\!\left(z\right) = C_{\ell}\!\left(\eta\right) {z}^{\ell + 1} {e}^{\omega i z} \,{}_1F_1\!\left(1 + \ell + \omega i \eta, 2 \ell + 2, -2 \omega i z\right)

\omega \in \left\{-1, 1\right\} \,\mathbin{\operatorname{and}}\, \ell \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \eta \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(1 + \ell + i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, 1 + \ell - i \eta \notin \{0, -1, \ldots\}\right) \,\mathbin{\operatorname{and}}\, 2 \ell + 2 \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}
Definitions:
Fungrim symbol Notation Short description
CoulombFF,η ⁣(z)F_{\ell,\eta}\!\left(z\right) Regular Coulomb wave function
CoulombCC ⁣(η)C_{\ell}\!\left(\eta\right) Coulomb wave function Gamow factor
Powab{a}^{b} Power
Expez{e}^{z} Exponential function
ConstIii Imaginary unit
Hypergeometric1F11F1 ⁣(a,b,z)\,{}_1F_1\!\left(a, b, z\right) Kummer confluent hypergeometric function
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
Entry(ID("d280c5"),
    Formula(Equal(CoulombF(ell, eta, z), Mul(Mul(Mul(CoulombC(ell, eta), Pow(z, Add(ell, 1))), Exp(Mul(Mul(omega, ConstI), z))), Hypergeometric1F1(Add(Add(1, ell), Mul(Mul(omega, ConstI), eta)), Add(Mul(2, ell), 2), Neg(Mul(Mul(Mul(2, omega), ConstI), z)))))),
    Variables(omega, ell, eta, z),
    Assumptions(And(Element(omega, Set(-1, 1)), Element(ell, CC), Element(eta, CC), And(NotElement(Add(Add(1, ell), Mul(ConstI, eta)), ZZLessEqual(0)), NotElement(Sub(Add(1, ell), Mul(ConstI, eta)), ZZLessEqual(0))), NotElement(Add(Mul(2, ell), 2), ZZLessEqual(0)), 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.

2019-10-05 13:11:19.856591 UTC