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Fungrim entry: 07a654

f(z)=(2ηz+(+1)z21)f(z)   where f(z)=c1F,η ⁣(z)+c2G,η ⁣(z)f''(z) = \left(\frac{2 \eta}{z} + \frac{\ell \left(\ell + 1\right)}{{z}^{2}} - 1\right) f(z)\; \text{ where } f(z) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)
Assumptions:c1C  and  c2C  and  C  and  ηC  and  (1++iη{0,1,}  and  1+iη{0,1,})  and  zC(,0]{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]
f''(z) = \left(\frac{2 \eta}{z} + \frac{\ell \left(\ell + 1\right)}{{z}^{2}} - 1\right) f(z)\; \text{ where } f(z) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)

{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]
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:
    Formula(Where(Equal(ComplexDerivative(f(z), For(z, z, 2)), Mul(Sub(Add(Div(Mul(2, eta), z), Div(Mul(ell, Add(ell, 1)), Pow(z, 2))), 1), f(z))), Equal(f(z), Add(Mul(Subscript(c, 1), CoulombF(ell, eta, z)), Mul(Subscript(c, 2), CoulombG(ell, eta, z)))))),
    Variables(ell, eta, Subscript(c, 1), Subscript(c, 2), z),
    Assumptions(And(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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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC