# Fungrim entry: 2fec14

$\frac{d}{d z}\, G_{\ell,\eta}\!\left(z\right) = \left(\frac{\ell + 1}{z} + \frac{\eta}{\ell + 1}\right) G_{\ell,\eta}\!\left(z\right) - \frac{\sqrt{1 + \ell + i \eta} \sqrt{1 + \ell - i \eta}}{\ell + 1} G_{\ell + 1,\eta}\!\left(z\right)$
Assumptions:$\ell \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \ell \ne -1 \,\mathbin{\operatorname{and}}\, \eta \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(1 + \ell + i \eta \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, 1 + \ell - i \eta \notin \{0, -1, \ldots\}\right) \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left(-\infty, 0\right]$
TeX:
\frac{d}{d z}\, G_{\ell,\eta}\!\left(z\right) = \left(\frac{\ell + 1}{z} + \frac{\eta}{\ell + 1}\right) G_{\ell,\eta}\!\left(z\right) - \frac{\sqrt{1 + \ell + i \eta} \sqrt{1 + \ell - i \eta}}{\ell + 1} G_{\ell + 1,\eta}\!\left(z\right)

\ell \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \ell \ne -1 \,\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
CoulombG$G_{\ell,\eta}\!\left(z\right)$ Irregular Coulomb wave function
Sqrt$\sqrt{z}$ Principal square root
ConstI$i$ Imaginary unit
CC$\mathbb{C}$ Complex numbers
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("2fec14"),
Assumptions(And(Element(ell, CC), Unequal(ell, -1), 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))))))