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Fungrim entry: eca10b

f(r+4)(z)(r+4)!=1z2(r2+7r+12)(2(r2+5r+6)zf(r+3)(z)(r+3)!+(r2+3r+z22zη(+1)+2)f(r+2)(z)(r+2)!+2(zη)f(r+1)(z)(r+1)!+f(r)(z)r!)   where f ⁣(z)=c1F,η ⁣(z)+c2G,η ⁣(z)\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\!\left(z\right) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)
Assumptions:rZ0andc1Candc2CandCandηCand(1++iη{0,1,}and1+iη{0,1,})andzC(,0]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\!\left(z\right) = {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
Derivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Derivative
Factorialn!n ! Factorial
Powab{a}^{b} Power
CoulombFF,η ⁣(z)F_{\ell,\eta}\!\left(z\right) Regular Coulomb wave function
CoulombGG,η ⁣(z)G_{\ell,\eta}\!\left(z\right) Irregular Coulomb wave function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
ConstIii Imaginary unit
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
Source code for this entry:
Entry(ID("eca10b"),
    Formula(Where(Equal(Div(Derivative(f(z), Tuple(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(Derivative(f(z), Tuple(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(Derivative(f(z), Tuple(z, z, Add(r, 2))), Factorial(Add(r, 2))))), Mul(Mul(2, Sub(z, eta)), Div(Derivative(f(z), Tuple(z, z, Add(r, 1))), Factorial(Add(r, 1))))), Div(Derivative(f(z), Tuple(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)),
    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))))))

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2019-08-21 11:44:15.926409 UTC