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Fungrim entry: 1976e1

F,η ⁣(z)=2z+1exp ⁣(logΓ ⁣(u)+logΓ ⁣(v)πη2)(eizU ⁣(u,2+2,2iz)(2iz)uΓ ⁣(v)+eizU ⁣(v,2+2,2iz)(2iz)vΓ ⁣(u))   where u=1++iη,v=1+iηF_{\ell,\eta}\!\left(z\right) = {2}^{\ell} {z}^{\ell + 1} \exp\!\left(\frac{\log \Gamma\!\left(u\right) + \log \Gamma\!\left(v\right) - \pi \eta}{2}\right) \left(\frac{{e}^{i z} U^{*}\!\left(u, 2 \ell + 2, -2 i z\right)}{{\left(2 i z\right)}^{u} \Gamma\!\left(v\right)} + \frac{{e}^{-i z} U^{*}\!\left(v, 2 \ell + 2, 2 i z\right)}{{\left(-2 i z\right)}^{v} \Gamma\!\left(u\right)}\right)\; \text{ where } u = 1 + \ell + i \eta,\,v = 1 + \ell - i \eta
Assumptions:CandηCand(1++iη{0,1,}and1+iη{0,1,})andzC{0}\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\{0\right\}
TeX:
F_{\ell,\eta}\!\left(z\right) = {2}^{\ell} {z}^{\ell + 1} \exp\!\left(\frac{\log \Gamma\!\left(u\right) + \log \Gamma\!\left(v\right) - \pi \eta}{2}\right) \left(\frac{{e}^{i z} U^{*}\!\left(u, 2 \ell + 2, -2 i z\right)}{{\left(2 i z\right)}^{u} \Gamma\!\left(v\right)} + \frac{{e}^{-i z} U^{*}\!\left(v, 2 \ell + 2, 2 i z\right)}{{\left(-2 i z\right)}^{v} \Gamma\!\left(u\right)}\right)\; \text{ where } u = 1 + \ell + i \eta,\,v = 1 + \ell - i \eta

\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\{0\right\}
Definitions:
Fungrim symbol Notation Short description
CoulombFF,η ⁣(z)F_{\ell,\eta}\!\left(z\right) Regular Coulomb wave function
Powab{a}^{b} Power
Expez{e}^{z} Exponential function
LogGammalogΓ ⁣(z)\log \Gamma\!\left(z\right) Logarithmic gamma function
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
HypergeometricUStarU ⁣(a,b,z)U^{*}\!\left(a, b, z\right) Scaled Tricomi confluent hypergeometric function
GammaFunctionΓ ⁣(z)\Gamma\!\left(z\right) Gamma function
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
Entry(ID("1976e1"),
    Formula(Equal(CoulombF(ell, eta, z), Where(Mul(Mul(Mul(Pow(2, ell), Pow(z, Add(ell, 1))), Exp(Div(Sub(Add(LogGamma(u), LogGamma(v)), Mul(ConstPi, eta)), 2))), Add(Div(Mul(Exp(Mul(ConstI, z)), HypergeometricUStar(u, Add(Mul(2, ell), 2), Neg(Mul(Mul(2, ConstI), z)))), Mul(Pow(Mul(Mul(2, ConstI), z), u), GammaFunction(v))), Div(Mul(Exp(Mul(Neg(ConstI), z)), HypergeometricUStar(v, Add(Mul(2, ell), 2), Mul(Mul(2, ConstI), z))), Mul(Pow(Neg(Mul(Mul(2, ConstI), z)), v), GammaFunction(u))))), Equal(u, Add(Add(1, ell), Mul(ConstI, eta))), Equal(v, Sub(Add(1, ell), Mul(ConstI, eta)))))),
    Variables(ell, eta, z),
    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, Set(0))))))

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

2019-09-20 18:07:53.062439 UTC