Fungrim home page

Fungrim entry: 304559

H,ηω ⁣(z)=F,η ⁣(z)eωiχF1,η ⁣(z)sin ⁣(χ)   where χ=σ ⁣(η)σ1 ⁣(η)(+12)πH^{\omega}_{\ell,\eta}\!\left(z\right) = \frac{F_{\ell,\eta}\!\left(z\right) {e}^{\omega i \chi} - 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:ω{1,1}andCandηCand2Zand1++iη{0,1,}and1+iη{0,1,}and+iη{0,1,}andiη{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}}\, 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:
H^{\omega}_{\ell,\eta}\!\left(z\right) = \frac{F_{\ell,\eta}\!\left(z\right) {e}^{\omega i \chi} - 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

\omega \in \left\{-1, 1\right\} \,\mathbin{\operatorname{and}}\, \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
CoulombHH,ηω ⁣(z)H^{\omega}_{\ell,\eta}\!\left(z\right) Outgoing and ingoing Coulomb wave function
CoulombFF,η ⁣(z)F_{\ell,\eta}\!\left(z\right) Regular Coulomb wave function
Expez{e}^{z} Exponential function
ConstIii Imaginary unit
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
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
Entry(ID("304559"),
    Formula(Where(Equal(CoulombH(omega, ell, eta, z), Div(Sub(Mul(CoulombF(ell, eta, z), Exp(Mul(Mul(omega, ConstI), 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(omega, ell, eta, z),
    Assumptions(And(Element(omega, Set(-1, 1)), 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))))))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-21 11:44:15.926409 UTC