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Fungrim entry: ad8df6

y(z)+(12ηz(+1)z2)y(z)=0   where y(z)=c1F,η ⁣(z)+c2G,η ⁣(z)y''(z) + \left(1 - \frac{2 \eta}{z} - \frac{\ell \left(\ell + 1\right)}{{z}^{2}}\right) y(z) = 0\; \text{ where } y(z) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)
Assumptions:C  and  ηC  and  (1++iη{0,1,}  and  1+iη{0,1,})  and  zC(,0]  and  c1C  and  c2C\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] \;\mathbin{\operatorname{and}}\; {c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C}
TeX:
y''(z) + \left(1 - \frac{2 \eta}{z} - \frac{\ell \left(\ell + 1\right)}{{z}^{2}}\right) y(z) = 0\; \text{ where } y(z) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)

\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] \;\mathbin{\operatorname{and}}\; {c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
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
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("ad8df6"),
    Formula(Where(Equal(Add(ComplexDerivative(y(z), For(z, z, 2)), Mul(Sub(Sub(1, Div(Mul(2, eta), z)), Div(Mul(ell, Add(ell, 1)), Pow(z, 2))), y(z))), 0), Equal(y(z), Add(Mul(Subscript(c, 1), CoulombF(ell, eta, z)), Mul(Subscript(c, 2), CoulombG(ell, eta, z)))))),
    Variables(ell, eta, z, Subscript(c, 1), Subscript(c, 2)),
    Assumptions(And(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))), Element(Subscript(c, 1), CC), Element(Subscript(c, 2), CC))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-04-08 16:14:44.404316 UTC