Fungrim entry: 01af55

H^{\omega}_{\ell,\eta}\!\left(z\right) = G_{\ell,\eta}\!\left(z\right) + \omega i F_{\ell,\eta}\!\left(z\right)

Assumptions:

\omega \in \left\{-1, 1\right\} \;\mathbin{\operatorname{and}}\; \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:

H^{\omega}_{\ell,\eta}\!\left(z\right) = G_{\ell,\eta}\!\left(z\right) + \omega i F_{\ell,\eta}\!\left(z\right)

\omega \in \left\{-1, 1\right\} \;\mathbin{\operatorname{and}}\; \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
`CoulombH`	$H^{\omega}_{\ell,\eta}\!\left(z\right)$	Outgoing and ingoing Coulomb wave function
`CoulombG`	$G_{\ell,\eta}\!\left(z\right)$	Irregular Coulomb wave function
`ConstI`	$i$	Imaginary unit
`CoulombF`	$F_{\ell,\eta}\!\left(z\right)$	Regular Coulomb wave function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("01af55"),
    Formula(Equal(CoulombH(omega, ell, eta, z), Add(CoulombG(ell, eta, z), Mul(Mul(omega, ConstI), CoulombF(ell, eta, z))))),
    Variables(omega, ell, eta, z),
    Assumptions(And(Element(omega, Set(-1, 1)), 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

Coulomb wave functions