# Fungrim entry: e20938

$G_{\ell,\eta}\!\left(z\right) = \frac{F_{\ell,\eta}\!\left(z\right) \cos(\chi) - F_{-\ell - 1,\eta}\!\left(z\right)}{\sin(\chi)}\; \text{ where } \chi = \sigma_{\ell}\!\left(\eta\right) - \sigma_{-\ell - 1}\!\left(\eta\right) - \left(\ell + \frac{1}{2}\right) \pi$
Assumptions:$\ell \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \eta \in \mathbb{C} \;\mathbin{\operatorname{and}}\; 2 \ell \notin \mathbb{Z} \;\mathbin{\operatorname{and}}\; 1 + \ell + i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; 1 + \ell - i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; -\ell + i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; -\ell - i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}$
TeX:
G_{\ell,\eta}\!\left(z\right) = \frac{F_{\ell,\eta}\!\left(z\right) \cos(\chi) - F_{-\ell - 1,\eta}\!\left(z\right)}{\sin(\chi)}\; \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
CoulombG$G_{\ell,\eta}\!\left(z\right)$ Irregular Coulomb wave function
CoulombF$F_{\ell,\eta}\!\left(z\right)$ Regular Coulomb wave function
Cos$\cos(z)$ Cosine
Sin$\sin(z)$ Sine
CoulombSigma$\sigma_{\ell}\!\left(\eta\right)$ Coulomb wave function phase shift
Pi$\pi$ The constant pi (3.14...)
CC$\mathbb{C}$ Complex numbers
ZZ$\mathbb{Z}$ Integers
ConstI$i$ Imaginary unit
ZZLessEqual$\mathbb{Z}_{\le n}$ 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)), Pi))))),
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))))))

## 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