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

Coulomb wave functions