Fungrim entry: 1976e1

F_{\ell,\eta}\!\left(z\right) = {2}^{\ell} {z}^{\ell + 1} \exp\!\left(\frac{\log \Gamma\!\left(u\right) + \log \Gamma\!\left(v\right) - \pi \eta}{2}\right) \left(\frac{{e}^{i z} U^{*}\!\left(u, 2 \ell + 2, -2 i z\right)}{{\left(2 i z\right)}^{u} \Gamma\!\left(v\right)} + \frac{{e}^{-i z} U^{*}\!\left(v, 2 \ell + 2, 2 i z\right)}{{\left(-2 i z\right)}^{v} \Gamma\!\left(u\right)}\right)\; \text{ where } u = 1 + \ell + i \eta,\,v = 1 + \ell - i \eta

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) \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

F_{\ell,\eta}\!\left(z\right) = {2}^{\ell} {z}^{\ell + 1} \exp\!\left(\frac{\log \Gamma\!\left(u\right) + \log \Gamma\!\left(v\right) - \pi \eta}{2}\right) \left(\frac{{e}^{i z} U^{*}\!\left(u, 2 \ell + 2, -2 i z\right)}{{\left(2 i z\right)}^{u} \Gamma\!\left(v\right)} + \frac{{e}^{-i z} U^{*}\!\left(v, 2 \ell + 2, 2 i z\right)}{{\left(-2 i z\right)}^{v} \Gamma\!\left(u\right)}\right)\; \text{ where } u = 1 + \ell + i \eta,\,v = 1 + \ell - i \eta

\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`	$\log \Gamma\!\left(z\right)$	Logarithmic gamma function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma 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("1976e1"),
    Formula(Equal(CoulombF(ell, eta, z), Where(Mul(Mul(Mul(Pow(2, ell), Pow(z, Add(ell, 1))), Exp(Div(Sub(Add(LogGamma(u), LogGamma(v)), Mul(ConstPi, eta)), 2))), Add(Div(Mul(Exp(Mul(ConstI, z)), HypergeometricUStar(u, Add(Mul(2, ell), 2), Neg(Mul(Mul(2, ConstI), z)))), Mul(Pow(Mul(Mul(2, ConstI), z), u), GammaFunction(v))), Div(Mul(Exp(Mul(Neg(ConstI), z)), HypergeometricUStar(v, Add(Mul(2, ell), 2), Mul(Mul(2, ConstI), z))), Mul(Pow(Neg(Mul(Mul(2, ConstI), z)), v), GammaFunction(u))))), Equal(u, Add(Add(1, ell), Mul(ConstI, eta))), Equal(v, Sub(Add(1, ell), Mul(ConstI, eta)))))),
    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))))))

Topics using this entry

Coulomb wave functions