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Fungrim entry: 01af55

H,ηω ⁣(z)=G,η ⁣(z)+ωiF,η ⁣(z)H^{\omega}_{\ell,\eta}\!\left(z\right) = G_{\ell,\eta}\!\left(z\right) + \omega i F_{\ell,\eta}\!\left(z\right)
Assumptions:ω{1,1}andCandηCand(1++iη{0,1,}and1+iη{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}}\, z \in \mathbb{C} \setminus \left\{0\right\}
TeX:
H^{\omega}_{\ell,\eta}\!\left(z\right) = G_{\ell,\eta}\!\left(z\right) + \omega i F_{\ell,\eta}\!\left(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}}\, z \in \mathbb{C} \setminus \left\{0\right\}
Definitions:
Fungrim symbol Notation Short description
CoulombHH,ηω ⁣(z)H^{\omega}_{\ell,\eta}\!\left(z\right) Outgoing and ingoing Coulomb wave function
CoulombGG,η ⁣(z)G_{\ell,\eta}\!\left(z\right) Irregular Coulomb wave function
ConstIii Imaginary unit
CoulombFF,η ⁣(z)F_{\ell,\eta}\!\left(z\right) Regular Coulomb wave 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("01af55"),
    Formula(Equal(CoulombH(omega, ell, eta, z), Add(CoulombG(ell, eta, z), Mul(Mul(omega, ConstI), CoulombF(ell, eta, 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))), 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