Fungrim entry: 781eae

H^{-}_{\ell,\eta}\!\left(z\right) = {\left(2 z\right)}^{i \eta} \exp\!\left(-i \left(z - \frac{\ell \pi}{2} + \sigma_{\ell}\!\left(\eta\right)\right)\right) U^{*}\!\left(1 + \ell - i \eta, 2 \ell + 2, 2 i z\right)

Assumptions:

\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\} \;\mathbin{\operatorname{and}}\; \left(\operatorname{Im}(z) < 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) \ge 0\right)

TeX:

H^{-}_{\ell,\eta}\!\left(z\right) = {\left(2 z\right)}^{i \eta} \exp\!\left(-i \left(z - \frac{\ell \pi}{2} + \sigma_{\ell}\!\left(\eta\right)\right)\right) U^{*}\!\left(1 + \ell - i \eta, 2 \ell + 2, 2 i z\right)

\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\} \;\mathbin{\operatorname{and}}\; \left(\operatorname{Im}(z) < 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) \ge 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`CoulombH`	$H^{\omega}_{\ell,\eta}\!\left(z\right)$	Outgoing and ingoing Coulomb wave function
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`CoulombSigma`	$\sigma_{\ell}\!\left(\eta\right)$	Coulomb wave function phase shift
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("781eae"),
    Formula(Equal(CoulombH(-1, ell, eta, z), Mul(Mul(Pow(Mul(2, z), Mul(ConstI, eta)), Exp(Neg(Mul(ConstI, Add(Sub(z, Div(Mul(ell, Pi), 2)), CoulombSigma(ell, eta)))))), HypergeometricUStar(Sub(Add(1, ell), Mul(ConstI, eta)), Add(Mul(2, ell), 2), Mul(Mul(2, ConstI), z))))),
    Variables(ell, eta, z),
    Assumptions(And(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))), Or(Less(Im(z), 0), GreaterEqual(Re(z), 0)))))

Topics using this entry

Coulomb wave functions