Fungrim entry: 4a4739

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:

\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
`CoulombC`	$C_{\ell}\!\left(\eta\right)$	Coulomb wave function Gamow factor
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`Exp`	${e}^{z}$	Exponential function
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`ConstI`	$i$	Imaginary unit
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	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), Gamma(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(Pi, 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

Coulomb wave functions