Fungrim entry: 2a2f18

F_{\ell,\eta}\!\left(z\right) = {2}^{\ell} \exp\!\left(\frac{\log \Gamma\!\left(1 + \ell + i \eta\right) + \log \Gamma\!\left(1 + \ell - i \eta\right) - \pi \eta}{2}\right) {z}^{\ell + 1} {e}^{\omega i z} \,{}_1{\textbf F}_1\!\left(1 + \ell + \omega i \eta, 2 \ell + 2, -2 \omega i z\right)

Assumptions:

\omega \in \left\{-1, 1\right\} \;\mathbin{\operatorname{and}}\; \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\}

TeX:

F_{\ell,\eta}\!\left(z\right) = {2}^{\ell} \exp\!\left(\frac{\log \Gamma\!\left(1 + \ell + i \eta\right) + \log \Gamma\!\left(1 + \ell - i \eta\right) - \pi \eta}{2}\right) {z}^{\ell + 1} {e}^{\omega i z} \,{}_1{\textbf F}_1\!\left(1 + \ell + \omega i \eta, 2 \ell + 2, -2 \omega i z\right)

\omega \in \left\{-1, 1\right\} \;\mathbin{\operatorname{and}}\; \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\}

Definitions:

Fungrim symbol	Notation	Short description
`CoulombF`	$F_{\ell,\eta}\!\left(z\right)$	Regular Coulomb wave function
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`ConstI`	$i$	Imaginary unit
`Pi`	$\pi$	The constant pi (3.14...)
`Hypergeometric1F1Regularized`	$\,{}_1{\textbf F}_1\!\left(a, b, z\right)$	Regularized Kummer confluent hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("2a2f18"),
    Formula(Equal(CoulombF(ell, eta, z), Mul(Mul(Mul(Mul(Pow(2, ell), Exp(Div(Sub(Add(LogGamma(Add(Add(1, ell), Mul(ConstI, eta))), LogGamma(Sub(Add(1, ell), Mul(ConstI, eta)))), Mul(Pi, eta)), 2))), Pow(z, Add(ell, 1))), Exp(Mul(Mul(omega, ConstI), z))), Hypergeometric1F1Regularized(Add(Add(1, ell), Mul(Mul(omega, ConstI), eta)), Add(Mul(2, ell), 2), Neg(Mul(Mul(Mul(2, omega), ConstI), z)))))),
    Variables(omega, ell, eta, z),
    Assumptions(And(Element(omega, Set(-1, 1)), 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))))))

Topics using this entry

Coulomb wave functions