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Fungrim entry: e20938

G,η ⁣(z)=F,η ⁣(z)cos ⁣(χ)F1,η ⁣(z)sin ⁣(χ)   where χ=σ ⁣(η)σ1 ⁣(η)(+12)πG_{\ell,\eta}\!\left(z\right) = \frac{F_{\ell,\eta}\!\left(z\right) \cos\!\left(\chi\right) - F_{-\ell - 1,\eta}\!\left(z\right)}{\sin\!\left(\chi\right)}\; \text{ where } \chi = \sigma_{\ell}\!\left(\eta\right) - \sigma_{-\ell - 1}\!\left(\eta\right) - \left(\ell + \frac{1}{2}\right) \pi
Assumptions:CandηCand2Zand1++iη{0,1,}and1+iη{0,1,}and+iη{0,1,}andiη{0,1,}andzC{0}\ell \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \eta \in \mathbb{C} \,\mathbin{\operatorname{and}}\, 2 \ell \notin \mathbb{Z} \,\mathbin{\operatorname{and}}\, 1 + \ell + i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, 1 + \ell - i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, -\ell + i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, -\ell - i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}
TeX:
G_{\ell,\eta}\!\left(z\right) = \frac{F_{\ell,\eta}\!\left(z\right) \cos\!\left(\chi\right) - F_{-\ell - 1,\eta}\!\left(z\right)}{\sin\!\left(\chi\right)}\; \text{ where } \chi = \sigma_{\ell}\!\left(\eta\right) - \sigma_{-\ell - 1}\!\left(\eta\right) - \left(\ell + \frac{1}{2}\right) \pi

\ell \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \eta \in \mathbb{C} \,\mathbin{\operatorname{and}}\, 2 \ell \notin \mathbb{Z} \,\mathbin{\operatorname{and}}\, 1 + \ell + i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, 1 + \ell - i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, -\ell + i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, -\ell - i \eta \notin \{0, -1, \ldots\} \,\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
CoulombFF,η ⁣(z)F_{\ell,\eta}\!\left(z\right) Regular Coulomb wave function
Sinsin ⁣(z)\sin\!\left(z\right) Sine
CoulombSigmaσ ⁣(η)\sigma_{\ell}\!\left(\eta\right) Coulomb wave function phase shift
ConstPiπ\pi The constant pi (3.14...)
CCC\mathbb{C} Complex numbers
ZZZ\mathbb{Z} Integers
ConstIii Imaginary unit
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
Entry(ID("e20938"),
    Formula(Where(Equal(CoulombG(ell, eta, z), Div(Sub(Mul(CoulombF(ell, eta, z), Cos(chi)), CoulombF(Sub(Neg(ell), 1), eta, z)), Sin(chi))), Equal(chi, Sub(Sub(CoulombSigma(ell, eta), CoulombSigma(Sub(Neg(ell), 1), eta)), Mul(Add(ell, Div(1, 2)), ConstPi))))),
    Variables(ell, eta, z),
    Assumptions(And(Element(ell, CC), Element(eta, CC), NotElement(Mul(2, ell), ZZ), NotElement(Add(Add(1, ell), Mul(ConstI, eta)), ZZLessEqual(0)), NotElement(Sub(Add(1, ell), Mul(ConstI, eta)), ZZLessEqual(0)), NotElement(Add(Neg(ell), Mul(ConstI, eta)), ZZLessEqual(0)), NotElement(Sub(Neg(ell), Mul(ConstI, eta)), ZZLessEqual(0)), Element(z, SetMinus(CC, Set(0))))))

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2019-08-21 11:44:15.926409 UTC