# Fungrim entry: 192a3e

$F_{\ell,\eta}\!\left(z\right) = \frac{H^{+}_{\ell,\eta}\!\left(z\right) - H^{-}_{\ell,\eta}\!\left(z\right)}{2 i}$
Assumptions:$\ell \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \eta \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(1 + \ell + i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; 1 + \ell - i \eta \notin \{0, -1, \ldots\}\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}$
TeX:
F_{\ell,\eta}\!\left(z\right) = \frac{H^{+}_{\ell,\eta}\!\left(z\right) - H^{-}_{\ell,\eta}\!\left(z\right)}{2 i}

\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
CoulombF$F_{\ell,\eta}\!\left(z\right)$ Regular Coulomb wave function
CoulombH$H^{\omega}_{\ell,\eta}\!\left(z\right)$ Outgoing and ingoing Coulomb wave function
ConstI$i$ Imaginary unit
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
Source code for this entry:
Entry(ID("192a3e"),
Formula(Equal(CoulombF(ell, eta, z), Div(Sub(CoulombH(1, ell, eta, z), CoulombH(-1, ell, eta, z)), Mul(2, ConstI)))),
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))))))

## Topics using this entry

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

2021-03-15 19:12:00.328586 UTC