# Fungrim entry: 74274a

$G_{\ell,\eta}\!\left(z\right) \left(\frac{d}{d z}\, F_{\ell,\eta}\!\left(z\right)\right) - \left(\frac{d}{d z}\, G_{\ell,\eta}\!\left(z\right)\right) F_{\ell,\eta}\!\left(z\right) = 1$
Assumptions:$\left(1 + \ell + i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, 1 + \ell - i \eta \notin \{0, -1, \ldots\}\right) \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left(-\infty, 0\right]$
TeX:
G_{\ell,\eta}\!\left(z\right) \left(\frac{d}{d z}\, F_{\ell,\eta}\!\left(z\right)\right) - \left(\frac{d}{d z}\, G_{\ell,\eta}\!\left(z\right)\right) F_{\ell,\eta}\!\left(z\right) = 1

\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(-\infty, 0\right]
Definitions:
Fungrim symbol Notation Short description
CoulombG$G_{\ell,\eta}\!\left(z\right)$ Irregular Coulomb wave function
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
CoulombF$F_{\ell,\eta}\!\left(z\right)$ Regular Coulomb wave function
ConstI$i$ Imaginary unit
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
CC$\mathbb{C}$ Complex numbers
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("74274a"),
Formula(Equal(Sub(Mul(CoulombG(ell, eta, z), Parentheses(ComplexDerivative(CoulombF(ell, eta, z), For(z, z, 1)))), Mul(Parentheses(ComplexDerivative(CoulombG(ell, eta, z), For(z, z, 1))), CoulombF(ell, eta, z))), 1)),
Variables(ell, eta, z),
Assumptions(And(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, OpenClosedInterval(Neg(Infinity), 0))))))

## Topics using this entry

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

2019-11-19 15:10:20.037976 UTC