# Fungrim entry: a51a4b

$\frac{d}{d z}\, F_{\ell,\eta}\!\left(z\right) = \left(\frac{\ell + 1}{z} + \frac{\eta}{\ell + 1}\right) F_{\ell,\eta}\!\left(z\right) - \frac{\sqrt{1 + \ell + i \eta} \sqrt{1 + \ell - i \eta}}{\ell + 1} F_{\ell + 1,\eta}\!\left(z\right)$
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}\, F_{\ell,\eta}\!\left(z\right) = \left(\frac{\ell + 1}{z} + \frac{\eta}{\ell + 1}\right) F_{\ell,\eta}\!\left(z\right) - \frac{\sqrt{1 + \ell + i \eta} \sqrt{1 + \ell - i \eta}}{\ell + 1} F_{\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
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
CoulombF$F_{\ell,\eta}\!\left(z\right)$ Regular 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("a51a4b"),
Assumptions(And(Element(ell, CC), NotEqual(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))))))