Fungrim entry: eca10b

\frac{{f}^{(r + 4)}(z)}{\left(r + 4\right)!} = -\frac{1}{{z}^{2} \left({r}^{2} + 7 r + 12\right)} \left(2 \left({r}^{2} + 5 r + 6\right) z \frac{{f}^{(r + 3)}(z)}{\left(r + 3\right)!} + \left({r}^{2} + 3 r + {z}^{2} - 2 z \eta - \ell \left(\ell + 1\right) + 2\right) \frac{{f}^{(r + 2)}(z)}{\left(r + 2\right)!} + 2 \left(z - \eta\right) \frac{{f}^{(r + 1)}(z)}{\left(r + 1\right)!} + \frac{{f}^{(r)}(z)}{r !}\right)\; \text{ where } f(z) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)

Assumptions:

r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; {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:

\frac{{f}^{(r + 4)}(z)}{\left(r + 4\right)!} = -\frac{1}{{z}^{2} \left({r}^{2} + 7 r + 12\right)} \left(2 \left({r}^{2} + 5 r + 6\right) z \frac{{f}^{(r + 3)}(z)}{\left(r + 3\right)!} + \left({r}^{2} + 3 r + {z}^{2} - 2 z \eta - \ell \left(\ell + 1\right) + 2\right) \frac{{f}^{(r + 2)}(z)}{\left(r + 2\right)!} + 2 \left(z - \eta\right) \frac{{f}^{(r + 1)}(z)}{\left(r + 1\right)!} + \frac{{f}^{(r)}(z)}{r !}\right)\; \text{ where } f(z) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)

r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; {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
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Factorial`	$n !$	Factorial
`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
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`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("eca10b"),
    Formula(Where(Equal(Div(ComplexDerivative(f(z), For(z, z, Add(r, 4))), Factorial(Add(r, 4))), Mul(Div(-1, Mul(Pow(z, 2), Add(Add(Pow(r, 2), Mul(7, r)), 12))), Add(Add(Add(Mul(Mul(Mul(2, Add(Add(Pow(r, 2), Mul(5, r)), 6)), z), Div(ComplexDerivative(f(z), For(z, z, Add(r, 3))), Factorial(Add(r, 3)))), Mul(Add(Sub(Sub(Add(Add(Pow(r, 2), Mul(3, r)), Pow(z, 2)), Mul(Mul(2, z), eta)), Mul(ell, Add(ell, 1))), 2), Div(ComplexDerivative(f(z), For(z, z, Add(r, 2))), Factorial(Add(r, 2))))), Mul(Mul(2, Sub(z, eta)), Div(ComplexDerivative(f(z), For(z, z, Add(r, 1))), Factorial(Add(r, 1))))), Div(ComplexDerivative(f(z), For(z, z, r)), Factorial(r))))), Equal(f(z), Add(Mul(Subscript(c, 1), CoulombF(ell, eta, z)), Mul(Subscript(c, 2), CoulombG(ell, eta, z)))))),
    Variables(r, ell, eta, Subscript(c, 1), Subscript(c, 2), z),
    Assumptions(And(Element(r, ZZGreaterEqual(0)), 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

Coulomb wave functions