Fungrim home page

# Fungrim entry: 304559

$H^{\omega}_{\ell,\eta}\!\left(z\right) = \frac{F_{\ell,\eta}\!\left(z\right) {e}^{\omega i \chi} - F_{-\ell - 1,\eta}\!\left(z\right)}{\sin\!\left(\chi\right)}\; \text{ where } \chi = \sigma_{\ell}\!\left(\eta\right) - \sigma_{-\ell - 1}\!\left(\eta\right) - \left(\ell + \frac{1}{2}\right) \pi$
Assumptions:$\omega \in \left\{-1, 1\right\} \,\mathbin{\operatorname{and}}\, \ell \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \eta \in \mathbb{C} \,\mathbin{\operatorname{and}}\, 2 \ell \notin \mathbb{Z} \,\mathbin{\operatorname{and}}\, 1 + \ell + i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, 1 + \ell - i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, -\ell + i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, -\ell - i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}$
TeX:
H^{\omega}_{\ell,\eta}\!\left(z\right) = \frac{F_{\ell,\eta}\!\left(z\right) {e}^{\omega i \chi} - F_{-\ell - 1,\eta}\!\left(z\right)}{\sin\!\left(\chi\right)}\; \text{ where } \chi = \sigma_{\ell}\!\left(\eta\right) - \sigma_{-\ell - 1}\!\left(\eta\right) - \left(\ell + \frac{1}{2}\right) \pi

\omega \in \left\{-1, 1\right\} \,\mathbin{\operatorname{and}}\, \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
CoulombH$H^{\omega}_{\ell,\eta}\!\left(z\right)$ Outgoing and ingoing Coulomb wave function
CoulombF$F_{\ell,\eta}\!\left(z\right)$ Regular Coulomb wave function
Exp${e}^{z}$ Exponential function
ConstI$i$ Imaginary unit
Sin$\sin\!\left(z\right)$ Sine
CoulombSigma$\sigma_{\ell}\!\left(\eta\right)$ Coulomb wave function phase shift
ConstPi$\pi$ The constant pi (3.14...)
CC$\mathbb{C}$ Complex numbers
ZZ$\mathbb{Z}$ Integers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
Source code for this entry:
Entry(ID("304559"),
Formula(Where(Equal(CoulombH(omega, ell, eta, z), Div(Sub(Mul(CoulombF(ell, eta, z), Exp(Mul(Mul(omega, ConstI), 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)), ConstPi))))),
Variables(omega, ell, eta, z),
Assumptions(And(Element(omega, Set(-1, 1)), 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

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

2019-06-18 07:49:59.356594 UTC