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Fungrim entry: 781eae

Hl,η ⁣(z)=(2z)iηexp ⁣(i(zπ2+σ ⁣(η)))U ⁣(1+iη,2+2,2iz)H^{-}_{l,\eta}\!\left(z\right) = {\left(2 z\right)}^{i \eta} \exp\!\left(-i \left(z - \frac{\ell \pi}{2} + \sigma_{\ell}\!\left(\eta\right)\right)\right) U^{*}\!\left(1 + \ell - i \eta, 2 \ell + 2, 2 i z\right)
Assumptions:CandηCand(1++iη{0,1,}and1+iη{0,1,})andzC{0}and(Im ⁣(z)<0orRe ⁣(z)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\} \,\mathbin{\operatorname{and}}\, \left(\operatorname{Im}\!\left(z\right) < 0 \,\mathbin{\operatorname{or}}\, \operatorname{Re}\!\left(z\right) \ge 0\right)
TeX:
H^{-}_{l,\eta}\!\left(z\right) = {\left(2 z\right)}^{i \eta} \exp\!\left(-i \left(z - \frac{\ell \pi}{2} + \sigma_{\ell}\!\left(\eta\right)\right)\right) U^{*}\!\left(1 + \ell - i \eta, 2 \ell + 2, 2 i 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\{0\right\} \,\mathbin{\operatorname{and}}\, \left(\operatorname{Im}\!\left(z\right) < 0 \,\mathbin{\operatorname{or}}\, \operatorname{Re}\!\left(z\right) \ge 0\right)
Definitions:
Fungrim symbol Notation Short description
CoulombHH,ηω ⁣(z)H^{\omega}_{\ell,\eta}\!\left(z\right) Outgoing and ingoing Coulomb wave function
Powab{a}^{b} Power
ConstIii Imaginary unit
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
CoulombSigmaσ ⁣(η)\sigma_{\ell}\!\left(\eta\right) Coulomb wave function phase shift
HypergeometricUStarU ⁣(a,b,z)U^{*}\!\left(a, b, z\right) Scaled Tricomi confluent hypergeometric function
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
ImIm ⁣(z)\operatorname{Im}\!\left(z\right) Imaginary part
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
Source code for this entry:
Entry(ID("781eae"),
    Formula(Equal(CoulombH(-1, l, eta, z), Mul(Mul(Pow(Mul(2, z), Mul(ConstI, eta)), Exp(Neg(Mul(ConstI, Add(Sub(z, Div(Mul(ell, ConstPi), 2)), CoulombSigma(ell, eta)))))), HypergeometricUStar(Sub(Add(1, ell), Mul(ConstI, eta)), Add(Mul(2, ell), 2), Mul(Mul(2, ConstI), z))))),
    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))), Or(Less(Im(z), 0), GreaterEqual(Re(z), 0)))))

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2019-09-16 21:17:18.797188 UTC