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Fungrim entry: bcdfc6

H,η+ ⁣(z)=(2z)iηexp ⁣(i(zπ2+σ ⁣(η)))U ⁣(1++iη,2+2,2iz)H^{+}_{\ell,\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:C  and  ηC  and  (1++iη{0,1,}  and  1+iη{0,1,})  and  zC{0}  and  (Im(z)0  or  Re(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}(z) \ge 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) > 0\right)
TeX:
H^{+}_{\ell,\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}(z) \ge 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) > 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
Piπ\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}(z) Imaginary part
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
Entry(ID("bcdfc6"),
    Formula(Equal(CoulombH(1, ell, eta, z), Mul(Mul(Pow(Mul(2, z), Neg(Mul(ConstI, eta))), Exp(Mul(ConstI, Add(Sub(z, Div(Mul(ell, Pi), 2)), CoulombSigma(ell, eta))))), HypergeometricUStar(Add(Add(1, ell), Mul(ConstI, eta)), Add(Mul(2, ell), 2), Neg(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(GreaterEqual(Im(z), 0), Greater(Re(z), 0)))))

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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