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Fungrim entry: 8547ab

G,η ⁣(z)=H,η+ ⁣(z)+H,η ⁣(z)2G_{\ell,\eta}\!\left(z\right) = \frac{H^{+}_{\ell,\eta}\!\left(z\right) + H^{-}_{\ell,\eta}\!\left(z\right)}{2}
Assumptions:CandηCand(1++iη{0,1,}and1+iη{0,1,})andzC{0}\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:
G_{\ell,\eta}\!\left(z\right) = \frac{H^{+}_{\ell,\eta}\!\left(z\right) + H^{-}_{\ell,\eta}\!\left(z\right)}{2}

\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
CoulombGG,η ⁣(z)G_{\ell,\eta}\!\left(z\right) Irregular Coulomb wave function
CoulombHH,ηω ⁣(z)H^{\omega}_{\ell,\eta}\!\left(z\right) Outgoing and ingoing Coulomb wave function
CCC\mathbb{C} Complex numbers
ConstIii Imaginary unit
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
Entry(ID("8547ab"),
    Formula(Equal(CoulombG(ell, eta, z), Div(Add(CoulombH(1, ell, eta, z), CoulombH(-1, ell, eta, z)), 2))),
    Variables(ell, eta, z),
    Assumptions(And(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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2019-08-21 11:44:15.926409 UTC