# Fungrim entry: 2a2f18

$F_{\ell,\eta}\!\left(z\right) = {2}^{\ell} \exp\!\left(\frac{\log \Gamma\!\left(1 + \ell + i \eta\right) + \log \Gamma\!\left(1 + \ell - i \eta\right) - \pi \eta}{2}\right) {z}^{\ell + 1} {e}^{\omega i z} \,{}_1{\textbf F}_1\!\left(1 + \ell + \omega i \eta, 2 \ell + 2, -2 \omega i z\right)$
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:
F_{\ell,\eta}\!\left(z\right) = {2}^{\ell} \exp\!\left(\frac{\log \Gamma\!\left(1 + \ell + i \eta\right) + \log \Gamma\!\left(1 + \ell - i \eta\right) - \pi \eta}{2}\right) {z}^{\ell + 1} {e}^{\omega i z} \,{}_1{\textbf F}_1\!\left(1 + \ell + \omega i \eta, 2 \ell + 2, -2 \omega i 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
CoulombF$F_{\ell,\eta}\!\left(z\right)$ Regular Coulomb wave function
Pow${a}^{b}$ Power
Exp${e}^{z}$ Exponential function
LogGamma$\log \Gamma(z)$ Logarithmic gamma function
ConstI$i$ Imaginary unit
Pi$\pi$ The constant pi (3.14...)
Hypergeometric1F1Regularized$\,{}_1{\textbf F}_1\!\left(a, b, z\right)$ Regularized Kummer confluent hypergeometric 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("2a2f18"),
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

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