# Fungrim entry: 0cc301

$H^{-}_{\ell,\eta}\!\left(z\right) = {\left(2 z\right)}^{-i \eta} \exp\!\left(i \left(z - \frac{\ell \pi}{2} + \sigma_{\ell}\!\left(\eta\right)\right)\right) U^{*}\!\left(1 + \ell + i \eta, 2 \ell + 2, -2 i z\right) - 2 i F_{\ell,\eta}\!\left(z\right)$
Assumptions:$\ell \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \eta \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(1 + \ell + i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; 1 + \ell - i \eta \notin \{0, -1, \ldots\}\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(\operatorname{Im}(z) \ge 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) > 0\right)$
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) - 2 i F_{\ell,\eta}\!\left(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) \ge 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) > 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
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
Im$\operatorname{Im}(z)$ Imaginary part
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("0cc301"),
Formula(Equal(CoulombH(-1, ell, eta, z), Sub(Mul(Mul(Pow(Mul(2, z), Neg(Mul(ConstI, eta))), Exp(Mul(ConstI, Add(Sub(z, Div(Mul(ell, Pi), 2)), CoulombSigma(ell, eta))))), HypergeometricUStar(Add(Add(1, ell), Mul(ConstI, eta)), Add(Mul(2, ell), 2), Neg(Mul(Mul(2, ConstI), z)))), Mul(Mul(2, ConstI), CoulombF(ell, eta, 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(GreaterEqual(Im(z), 0), Greater(Re(z), 0)))))

## Topics using this entry

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

2021-03-15 19:12:00.328586 UTC