Fungrim home page

Fungrim entry: 4a4739

C ⁣(η)=2Γ ⁣(2+2)exp ⁣(logΓ ⁣(1++iη)+logΓ ⁣(1+iη)πη2)C_{\ell}\!\left(\eta\right) = \frac{{2}^{\ell}}{\Gamma\!\left(2 \ell + 2\right)} \exp\!\left(\frac{\log \Gamma\!\left(1 + \ell + i \eta\right) + \log \Gamma\!\left(1 + \ell - i \eta\right) - \pi \eta}{2}\right)
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:
C_{\ell}\!\left(\eta\right) = \frac{{2}^{\ell}}{\Gamma\!\left(2 \ell + 2\right)} \exp\!\left(\frac{\log \Gamma\!\left(1 + \ell + i \eta\right) + \log \Gamma\!\left(1 + \ell - i \eta\right) - \pi \eta}{2}\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)
Definitions:
Fungrim symbol Notation Short description
CoulombCC ⁣(η)C_{\ell}\!\left(\eta\right) Coulomb wave function Gamow factor
Powab{a}^{b} Power
GammaFunctionΓ(z)\Gamma(z) Gamma function
Expez{e}^{z} Exponential function
LogGammalogΓ(z)\log \Gamma(z) Logarithmic gamma function
ConstIii Imaginary unit
ConstPiπ\pi The constant pi (3.14...)
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
Entry(ID("4a4739"),
    Formula(Equal(CoulombC(ell, eta), Mul(Div(Pow(2, ell), GammaFunction(Add(Mul(2, ell), 2))), Exp(Div(Sub(Add(LogGamma(Add(Add(1, ell), Mul(ConstI, eta))), LogGamma(Sub(Add(1, ell), Mul(ConstI, eta)))), Mul(ConstPi, eta)), 2))))),
    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))))))

Topics using this entry

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