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Fungrim entry: ed2bf6

σ ⁣(η)=logΓ ⁣(1++iη)logΓ ⁣(1+iη)2i\sigma_{\ell}\!\left(\eta\right) = \frac{\log \Gamma\!\left(1 + \ell + i \eta\right) - \log \Gamma\!\left(1 + \ell - i \eta\right)}{2 i}
Assumptions:CandηCand(1++iη{0,1,}and1+iη{0,1,})\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)
TeX:
\sigma_{\ell}\!\left(\eta\right) = \frac{\log \Gamma\!\left(1 + \ell + i \eta\right) - \log \Gamma\!\left(1 + \ell - i \eta\right)}{2 i}

\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)
Definitions:
Fungrim symbol Notation Short description
CoulombSigmaσ ⁣(η)\sigma_{\ell}\!\left(\eta\right) Coulomb wave function phase shift
LogGammalogΓ ⁣(z)\log \Gamma\!\left(z\right) Logarithmic gamma function
ConstIii Imaginary unit
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
Entry(ID("ed2bf6"),
    Formula(Equal(CoulombSigma(ell, eta), Div(Sub(LogGamma(Add(Add(1, ell), Mul(ConstI, eta))), LogGamma(Sub(Add(1, ell), Mul(ConstI, eta)))), Mul(2, ConstI)))),
    Variables(ell, eta),
    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))))))

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

2019-08-21 11:44:15.926409 UTC