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

Coulomb wave functions