# Fungrim entry: faa118

$f'''(z) = \left(\frac{2 \eta}{z} + \frac{\ell \left(\ell + 1\right)}{{z}^{2}} - 1\right) f'(z) - 2 \left(\frac{\eta}{{z}^{2}} + \frac{\ell \left(\ell + 1\right)}{{z}^{3}}\right) f\!\left(z\right)\; \text{ where } f\!\left(z\right) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)$
Assumptions:${c}_{1} \in \mathbb{C} \,\mathbin{\operatorname{and}}\, {c}_{2} \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \ell \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \eta \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(1 + \ell + i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, 1 + \ell - i \eta \notin \{0, -1, \ldots\}\right) \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left(-\infty, 0\right]$
TeX:
f'''(z) = \left(\frac{2 \eta}{z} + \frac{\ell \left(\ell + 1\right)}{{z}^{2}} - 1\right) f'(z) - 2 \left(\frac{\eta}{{z}^{2}} + \frac{\ell \left(\ell + 1\right)}{{z}^{3}}\right) f\!\left(z\right)\; \text{ where } f\!\left(z\right) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)

{c}_{1} \in \mathbb{C} \,\mathbin{\operatorname{and}}\, {c}_{2} \in \mathbb{C} \,\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(-\infty, 0\right]
Definitions:
Fungrim symbol Notation Short description
Derivative$\frac{d}{d z}\, f\!\left(z\right)$ Derivative
Pow${a}^{b}$ Power
CoulombF$F_{\ell,\eta}\!\left(z\right)$ Regular Coulomb wave function
CoulombG$G_{\ell,\eta}\!\left(z\right)$ Irregular Coulomb wave function
CC$\mathbb{C}$ Complex numbers
ConstI$i$ Imaginary unit
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("faa118"),
Formula(Where(Equal(Derivative(f(z), Tuple(z, z, 3)), Sub(Mul(Sub(Add(Div(Mul(2, eta), z), Div(Mul(ell, Add(ell, 1)), Pow(z, 2))), 1), Derivative(f(z), Tuple(z, z, 1))), Mul(Mul(2, Add(Div(eta, Pow(z, 2)), Div(Mul(ell, Add(ell, 1)), Pow(z, 3)))), f(z)))), Equal(f(z), Add(Mul(Subscript(c, 1), CoulombF(ell, eta, z)), Mul(Subscript(c, 2), CoulombG(ell, eta, z)))))),
Variables(ell, eta, Subscript(c, 1), Subscript(c, 2)),
Assumptions(And(Element(Subscript(c, 1), CC), Element(Subscript(c, 2), CC), 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, OpenClosedInterval(Neg(Infinity), 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