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Fungrim entry: a51a4b

ddzF,η ⁣(z)=(+1z+η+1)F,η ⁣(z)1++iη1+iη+1F+1,η ⁣(z)\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:Cand1andηCand(1++iη{0,1,}and1+iη{0,1,})andzC(,0]\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
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
CoulombFF,η ⁣(z)F_{\ell,\eta}\!\left(z\right) Regular Coulomb wave function
Sqrtz\sqrt{z} Principal square root
ConstIii Imaginary unit
CCC\mathbb{C} Complex numbers
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("a51a4b"),
    Formula(Equal(ComplexDerivative(CoulombF(ell, eta, z), For(z, z, 1)), Sub(Mul(Add(Div(Add(ell, 1), z), Div(eta, Add(ell, 1))), CoulombF(ell, eta, z)), Mul(Div(Mul(Sqrt(Add(Add(1, ell), Mul(ConstI, eta))), Sqrt(Sub(Add(1, ell), Mul(ConstI, eta)))), Add(ell, 1)), CoulombF(Add(ell, 1), eta, z))))),
    Variables(ell, eta),
    Assumptions(And(Element(ell, CC), Unequal(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))))))

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

2019-10-05 13:11:19.856591 UTC