Fungrim entry: e2efbf

G_{\ell,\eta}\!\left(z\right) = \frac{1}{2} \left({\left(2 z\right)}^{-i \eta} \exp\!\left(i \left(z - \frac{\ell \pi}{2} + \sigma_{\ell}\!\left(\eta\right)\right)\right) U^{*}\!\left(1 + \ell + i \eta, 2 \ell + 2, -2 i z\right) + {\left(2 z\right)}^{i \eta} \exp\!\left(-i \left(z - \frac{\ell \pi}{2} + \sigma_{\ell}\!\left(\eta\right)\right)\right) U^{*}\!\left(1 + \ell - i \eta, 2 \ell + 2, 2 i z\right)\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) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0

TeX:

G_{\ell,\eta}\!\left(z\right) = \frac{1}{2} \left({\left(2 z\right)}^{-i \eta} \exp\!\left(i \left(z - \frac{\ell \pi}{2} + \sigma_{\ell}\!\left(\eta\right)\right)\right) U^{*}\!\left(1 + \ell + i \eta, 2 \ell + 2, -2 i z\right) + {\left(2 z\right)}^{i \eta} \exp\!\left(-i \left(z - \frac{\ell \pi}{2} + \sigma_{\ell}\!\left(\eta\right)\right)\right) U^{*}\!\left(1 + \ell - i \eta, 2 \ell + 2, 2 i z\right)\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) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0

Definitions:

Fungrim symbol	Notation	Short description
`CoulombG`	$G_{\ell,\eta}\!\left(z\right)$	Irregular Coulomb wave function
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`CoulombSigma`	$\sigma_{\ell}\!\left(\eta\right)$	Coulomb wave function phase shift
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("e2efbf"),
    Formula(Equal(CoulombG(ell, eta, z), Mul(Div(1, 2), Add(Mul(Mul(Pow(Mul(2, z), Neg(Mul(ConstI, eta))), Exp(Mul(ConstI, Add(Sub(z, Div(Mul(ell, Pi), 2)), CoulombSigma(ell, eta))))), HypergeometricUStar(Add(Add(1, ell), Mul(ConstI, eta)), Add(Mul(2, ell), 2), Neg(Mul(Mul(2, ConstI), z)))), Mul(Mul(Pow(Mul(2, z), Mul(ConstI, eta)), Exp(Neg(Mul(ConstI, Add(Sub(z, Div(Mul(ell, Pi), 2)), CoulombSigma(ell, eta)))))), HypergeometricUStar(Sub(Add(1, ell), Mul(ConstI, eta)), Add(Mul(2, ell), 2), Mul(Mul(2, ConstI), z))))))),
    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))), Greater(Re(z), 0))))

Topics using this entry

Coulomb wave functions