Coulomb wave functions

Symbol: CoulombF —

F_{\ell,\eta}\!\left(z\right)

— Regular Coulomb wave function

Definitions:

Fungrim symbol	Notation	Short description
`CoulombF`	$F_{\ell,\eta}\!\left(z\right)$	Regular Coulomb wave function

Source code for this entry:

Entry(ID("8b2cb9"),
    SymbolDefinition(CoulombF, CoulombF(ell, eta, z), "Regular Coulomb wave function"))

f25e3d

Symbol: CoulombG —

G_{\ell,\eta}\!\left(z\right)

— Irregular Coulomb wave function

Definitions:

Fungrim symbol	Notation	Short description
`CoulombG`	$G_{\ell,\eta}\!\left(z\right)$	Irregular Coulomb wave function

Source code for this entry:

Entry(ID("f25e3d"),
    SymbolDefinition(CoulombG, CoulombG(ell, eta, z), "Irregular Coulomb wave function"))

16a4e7

Symbol: CoulombH —

H^{\omega}_{\ell,\eta}\!\left(z\right)

— Outgoing and ingoing Coulomb wave function

Definitions:

Fungrim symbol	Notation	Short description
`CoulombH`	$H^{\omega}_{\ell,\eta}\!\left(z\right)$	Outgoing and ingoing Coulomb wave function

Source code for this entry:

Entry(ID("16a4e7"),
    SymbolDefinition(CoulombH, CoulombH(omega, ell, eta, z), "Outgoing and ingoing Coulomb wave function"))

2b12f4

Symbol: CoulombC —

C_{\ell}\!\left(\eta\right)

— Coulomb wave function Gamow factor

Definitions:

Fungrim symbol	Notation	Short description
`CoulombC`	$C_{\ell}\!\left(\eta\right)$	Coulomb wave function Gamow factor

Source code for this entry:

Entry(ID("2b12f4"),
    SymbolDefinition(CoulombC, CoulombC(ell, eta), "Coulomb wave function Gamow factor"))

512063

Symbol: CoulombSigma —

\sigma_{\ell}\!\left(\eta\right)

— Coulomb wave function phase shift

Definitions:

Fungrim symbol	Notation	Short description
`CoulombSigma`	$\sigma_{\ell}\!\left(\eta\right)$	Coulomb wave function phase shift

Source code for this entry:

Entry(ID("512063"),
    SymbolDefinition(CoulombSigma, CoulombSigma(ell, eta), "Coulomb wave function phase shift"))

ad8df6

y''(z) + \left(1 - \frac{2 \eta}{z} - \frac{\ell \left(\ell + 1\right)}{{z}^{2}}\right) y(z) = 0\; \text{ where } y(z) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\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(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; {c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C}

TeX:

y''(z) + \left(1 - \frac{2 \eta}{z} - \frac{\ell \left(\ell + 1\right)}{{z}^{2}}\right) y(z) = 0\; \text{ where } y(z) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\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(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; {c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Pow`	${a}^{b}$	Power
`CoulombF`	$F_{\ell,\eta}\!\left(z\right)$	Regular Coulomb wave function
`CoulombG`	$G_{\ell,\eta}\!\left(z\right)$	Irregular Coulomb wave function
`CC`	$\mathbb{C}$	Complex numbers
`ConstI`	$i$	Imaginary unit
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("ad8df6"),
    Formula(Where(Equal(Add(ComplexDerivative(y(z), For(z, z, 2)), Mul(Sub(Sub(1, Div(Mul(2, eta), z)), Div(Mul(ell, Add(ell, 1)), Pow(z, 2))), y(z))), 0), Equal(y(z), Add(Mul(Subscript(c, 1), CoulombF(ell, eta, z)), Mul(Subscript(c, 2), CoulombG(ell, eta, z)))))),
    Variables(ell, eta, z, Subscript(c, 1), Subscript(c, 2)),
    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, OpenClosedInterval(Neg(Infinity), 0))), Element(Subscript(c, 1), CC), Element(Subscript(c, 2), CC))))

74274a

G_{\ell,\eta}\!\left(z\right) \left(\frac{d}{d z}\, F_{\ell,\eta}\!\left(z\right)\right) - \left(\frac{d}{d z}\, G_{\ell,\eta}\!\left(z\right)\right) F_{\ell,\eta}\!\left(z\right) = 1

Assumptions:

\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(-\infty, 0\right]

TeX:

G_{\ell,\eta}\!\left(z\right) \left(\frac{d}{d z}\, F_{\ell,\eta}\!\left(z\right)\right) - \left(\frac{d}{d z}\, G_{\ell,\eta}\!\left(z\right)\right) F_{\ell,\eta}\!\left(z\right) = 1

\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(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`CoulombG`	$G_{\ell,\eta}\!\left(z\right)$	Irregular Coulomb wave function
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`CoulombF`	$F_{\ell,\eta}\!\left(z\right)$	Regular Coulomb wave function
`ConstI`	$i$	Imaginary unit
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("74274a"),
    Formula(Equal(Sub(Mul(CoulombG(ell, eta, z), Parentheses(ComplexDerivative(CoulombF(ell, eta, z), For(z, z, 1)))), Mul(Parentheses(ComplexDerivative(CoulombG(ell, eta, z), For(z, z, 1))), CoulombF(ell, eta, z))), 1)),
    Variables(ell, eta, z),
    Assumptions(And(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, OpenClosedInterval(Neg(Infinity), 0))))))

192a3e

F_{\ell,\eta}\!\left(z\right) = \frac{H^{+}_{\ell,\eta}\!\left(z\right) - H^{-}_{\ell,\eta}\!\left(z\right)}{2 i}

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) = \frac{H^{+}_{\ell,\eta}\!\left(z\right) - H^{-}_{\ell,\eta}\!\left(z\right)}{2 i}

\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
`CoulombH`	$H^{\omega}_{\ell,\eta}\!\left(z\right)$	Outgoing and ingoing Coulomb wave function
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("192a3e"),
    Formula(Equal(CoulombF(ell, eta, z), Div(Sub(CoulombH(1, ell, eta, z), CoulombH(-1, ell, eta, z)), Mul(2, ConstI)))),
    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))))))

8547ab

G_{\ell,\eta}\!\left(z\right) = \frac{H^{+}_{\ell,\eta}\!\left(z\right) + H^{-}_{\ell,\eta}\!\left(z\right)}{2}

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:

G_{\ell,\eta}\!\left(z\right) = \frac{H^{+}_{\ell,\eta}\!\left(z\right) + H^{-}_{\ell,\eta}\!\left(z\right)}{2}

\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
`CoulombG`	$G_{\ell,\eta}\!\left(z\right)$	Irregular Coulomb wave function
`CoulombH`	$H^{\omega}_{\ell,\eta}\!\left(z\right)$	Outgoing and ingoing Coulomb wave function
`CC`	$\mathbb{C}$	Complex numbers
`ConstI`	$i$	Imaginary unit
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("8547ab"),
    Formula(Equal(CoulombG(ell, eta, z), Div(Add(CoulombH(1, ell, eta, z), CoulombH(-1, ell, eta, z)), 2))),
    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))))))

01af55

H^{\omega}_{\ell,\eta}\!\left(z\right) = G_{\ell,\eta}\!\left(z\right) + \omega i F_{\ell,\eta}\!\left(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:

H^{\omega}_{\ell,\eta}\!\left(z\right) = G_{\ell,\eta}\!\left(z\right) + \omega i F_{\ell,\eta}\!\left(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
`CoulombH`	$H^{\omega}_{\ell,\eta}\!\left(z\right)$	Outgoing and ingoing Coulomb wave function
`CoulombG`	$G_{\ell,\eta}\!\left(z\right)$	Irregular Coulomb wave function
`ConstI`	$i$	Imaginary unit
`CoulombF`	$F_{\ell,\eta}\!\left(z\right)$	Regular Coulomb wave 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("01af55"),
    Formula(Equal(CoulombH(omega, ell, eta, z), Add(CoulombG(ell, eta, z), Mul(Mul(omega, ConstI), CoulombF(ell, eta, 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))))))

e20938

G_{\ell,\eta}\!\left(z\right) = \frac{F_{\ell,\eta}\!\left(z\right) \cos(\chi) - F_{-\ell - 1,\eta}\!\left(z\right)}{\sin(\chi)}\; \text{ where } \chi = \sigma_{\ell}\!\left(\eta\right) - \sigma_{-\ell - 1}\!\left(\eta\right) - \left(\ell + \frac{1}{2}\right) \pi

Assumptions:

\ell \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \eta \in \mathbb{C} \;\mathbin{\operatorname{and}}\; 2 \ell \notin \mathbb{Z} \;\mathbin{\operatorname{and}}\; 1 + \ell + i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; 1 + \ell - i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; -\ell + i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; -\ell - i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

G_{\ell,\eta}\!\left(z\right) = \frac{F_{\ell,\eta}\!\left(z\right) \cos(\chi) - F_{-\ell - 1,\eta}\!\left(z\right)}{\sin(\chi)}\; \text{ where } \chi = \sigma_{\ell}\!\left(\eta\right) - \sigma_{-\ell - 1}\!\left(\eta\right) - \left(\ell + \frac{1}{2}\right) \pi

\ell \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \eta \in \mathbb{C} \;\mathbin{\operatorname{and}}\; 2 \ell \notin \mathbb{Z} \;\mathbin{\operatorname{and}}\; 1 + \ell + i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; 1 + \ell - i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; -\ell + i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; -\ell - i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`CoulombG`	$G_{\ell,\eta}\!\left(z\right)$	Irregular Coulomb wave function
`CoulombF`	$F_{\ell,\eta}\!\left(z\right)$	Regular Coulomb wave function
`Cos`	$\cos(z)$	Cosine
`Sin`	$\sin(z)$	Sine
`CoulombSigma`	$\sigma_{\ell}\!\left(\eta\right)$	Coulomb wave function phase shift
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers
`ConstI`	$i$	Imaginary unit
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("e20938"),
    Formula(Where(Equal(CoulombG(ell, eta, z), Div(Sub(Mul(CoulombF(ell, eta, z), Cos(chi)), CoulombF(Sub(Neg(ell), 1), eta, z)), Sin(chi))), Equal(chi, Sub(Sub(CoulombSigma(ell, eta), CoulombSigma(Sub(Neg(ell), 1), eta)), Mul(Add(ell, Div(1, 2)), Pi))))),
    Variables(ell, eta, z),
    Assumptions(And(Element(ell, CC), Element(eta, CC), NotElement(Mul(2, ell), ZZ), NotElement(Add(Add(1, ell), Mul(ConstI, eta)), ZZLessEqual(0)), NotElement(Sub(Add(1, ell), Mul(ConstI, eta)), ZZLessEqual(0)), NotElement(Add(Neg(ell), Mul(ConstI, eta)), ZZLessEqual(0)), NotElement(Sub(Neg(ell), Mul(ConstI, eta)), ZZLessEqual(0)), Element(z, SetMinus(CC, Set(0))))))

304559

H^{\omega}_{\ell,\eta}\!\left(z\right) = \frac{F_{\ell,\eta}\!\left(z\right) {e}^{\omega i \chi} - F_{-\ell - 1,\eta}\!\left(z\right)}{\sin(\chi)}\; \text{ where } \chi = \sigma_{\ell}\!\left(\eta\right) - \sigma_{-\ell - 1}\!\left(\eta\right) - \left(\ell + \frac{1}{2}\right) \pi

Assumptions:

\omega \in \left\{-1, 1\right\} \;\mathbin{\operatorname{and}}\; \ell \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \eta \in \mathbb{C} \;\mathbin{\operatorname{and}}\; 2 \ell \notin \mathbb{Z} \;\mathbin{\operatorname{and}}\; 1 + \ell + i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; 1 + \ell - i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; -\ell + i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; -\ell - i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

H^{\omega}_{\ell,\eta}\!\left(z\right) = \frac{F_{\ell,\eta}\!\left(z\right) {e}^{\omega i \chi} - F_{-\ell - 1,\eta}\!\left(z\right)}{\sin(\chi)}\; \text{ where } \chi = \sigma_{\ell}\!\left(\eta\right) - \sigma_{-\ell - 1}\!\left(\eta\right) - \left(\ell + \frac{1}{2}\right) \pi

\omega \in \left\{-1, 1\right\} \;\mathbin{\operatorname{and}}\; \ell \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \eta \in \mathbb{C} \;\mathbin{\operatorname{and}}\; 2 \ell \notin \mathbb{Z} \;\mathbin{\operatorname{and}}\; 1 + \ell + i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; 1 + \ell - i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; -\ell + i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; -\ell - i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`CoulombH`	$H^{\omega}_{\ell,\eta}\!\left(z\right)$	Outgoing and ingoing Coulomb wave function
`CoulombF`	$F_{\ell,\eta}\!\left(z\right)$	Regular Coulomb wave function
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`Sin`	$\sin(z)$	Sine
`CoulombSigma`	$\sigma_{\ell}\!\left(\eta\right)$	Coulomb wave function phase shift
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("304559"),
    Formula(Where(Equal(CoulombH(omega, ell, eta, z), Div(Sub(Mul(CoulombF(ell, eta, z), Exp(Mul(Mul(omega, ConstI), chi))), CoulombF(Sub(Neg(ell), 1), eta, z)), Sin(chi))), Equal(chi, Sub(Sub(CoulombSigma(ell, eta), CoulombSigma(Sub(Neg(ell), 1), eta)), Mul(Add(ell, Div(1, 2)), Pi))))),
    Variables(omega, ell, eta, z),
    Assumptions(And(Element(omega, Set(-1, 1)), Element(ell, CC), Element(eta, CC), NotElement(Mul(2, ell), ZZ), NotElement(Add(Add(1, ell), Mul(ConstI, eta)), ZZLessEqual(0)), NotElement(Sub(Add(1, ell), Mul(ConstI, eta)), ZZLessEqual(0)), NotElement(Add(Neg(ell), Mul(ConstI, eta)), ZZLessEqual(0)), NotElement(Sub(Neg(ell), Mul(ConstI, eta)), ZZLessEqual(0)), Element(z, SetMinus(CC, Set(0))))))

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))))))

ed2bf6

\sigma_{\ell}\!\left(\eta\right) = \frac{\log \Gamma\!\left(1 + \ell + i \eta\right) - \log \Gamma\!\left(1 + \ell - i \eta\right)}{2 i}

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:

\sigma_{\ell}\!\left(\eta\right) = \frac{\log \Gamma\!\left(1 + \ell + i \eta\right) - \log \Gamma\!\left(1 + \ell - i \eta\right)}{2 i}

\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
`CoulombSigma`	$\sigma_{\ell}\!\left(\eta\right)$	Coulomb wave function phase shift
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("ed2bf6"),
    Formula(Equal(CoulombSigma(ell, eta), Div(Sub(LogGamma(Add(Add(1, ell), Mul(ConstI, eta))), LogGamma(Sub(Add(1, ell), Mul(ConstI, eta)))), Mul(2, ConstI)))),
    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))))))

a51a4b

\frac{d}{d z}\, F_{\ell,\eta}\!\left(z\right) = \left(\frac{\ell + 1}{z} + \frac{\eta}{\ell + 1}\right) F_{\ell,\eta}\!\left(z\right) - \frac{\sqrt{1 + \ell + i \eta} \sqrt{1 + \ell - i \eta}}{\ell + 1} F_{\ell + 1,\eta}\!\left(z\right)

Assumptions:

\ell \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \ell \ne -1 \;\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(-\infty, 0\right]

TeX:

\frac{d}{d z}\, F_{\ell,\eta}\!\left(z\right) = \left(\frac{\ell + 1}{z} + \frac{\eta}{\ell + 1}\right) F_{\ell,\eta}\!\left(z\right) - \frac{\sqrt{1 + \ell + i \eta} \sqrt{1 + \ell - i \eta}}{\ell + 1} F_{\ell + 1,\eta}\!\left(z\right)

\ell \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \ell \ne -1 \;\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(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`CoulombF`	$F_{\ell,\eta}\!\left(z\right)$	Regular Coulomb wave function
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("a51a4b"),
    Formula(Equal(ComplexDerivative(CoulombF(ell, eta, z), For(z, z, 1)), Sub(Mul(Add(Div(Add(ell, 1), z), Div(eta, Add(ell, 1))), CoulombF(ell, eta, z)), Mul(Div(Mul(Sqrt(Add(Add(1, ell), Mul(ConstI, eta))), Sqrt(Sub(Add(1, ell), Mul(ConstI, eta)))), Add(ell, 1)), CoulombF(Add(ell, 1), eta, z))))),
    Variables(ell, eta, z),
    Assumptions(And(Element(ell, CC), NotEqual(ell, -1), 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, OpenClosedInterval(Neg(Infinity), 0))))))

2fec14

\frac{d}{d z}\, G_{\ell,\eta}\!\left(z\right) = \left(\frac{\ell + 1}{z} + \frac{\eta}{\ell + 1}\right) G_{\ell,\eta}\!\left(z\right) - \frac{\sqrt{1 + \ell + i \eta} \sqrt{1 + \ell - i \eta}}{\ell + 1} G_{\ell + 1,\eta}\!\left(z\right)

Assumptions:

\ell \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \ell \ne -1 \;\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(-\infty, 0\right]

TeX:

\frac{d}{d z}\, G_{\ell,\eta}\!\left(z\right) = \left(\frac{\ell + 1}{z} + \frac{\eta}{\ell + 1}\right) G_{\ell,\eta}\!\left(z\right) - \frac{\sqrt{1 + \ell + i \eta} \sqrt{1 + \ell - i \eta}}{\ell + 1} G_{\ell + 1,\eta}\!\left(z\right)

\ell \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \ell \ne -1 \;\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(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`CoulombG`	$G_{\ell,\eta}\!\left(z\right)$	Irregular Coulomb wave function
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("2fec14"),
    Formula(Equal(ComplexDerivative(CoulombG(ell, eta, z), For(z, z, 1)), Sub(Mul(Add(Div(Add(ell, 1), z), Div(eta, Add(ell, 1))), CoulombG(ell, eta, z)), Mul(Div(Mul(Sqrt(Add(Add(1, ell), Mul(ConstI, eta))), Sqrt(Sub(Add(1, ell), Mul(ConstI, eta)))), Add(ell, 1)), CoulombG(Add(ell, 1), eta, z))))),
    Variables(ell, eta, z),
    Assumptions(And(Element(ell, CC), NotEqual(ell, -1), 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, OpenClosedInterval(Neg(Infinity), 0))))))

07a654

f''(z) = \left(\frac{2 \eta}{z} + \frac{\ell \left(\ell + 1\right)}{{z}^{2}} - 1\right) f(z)\; \text{ where } f(z) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)

Assumptions:

{c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C} \;\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(-\infty, 0\right]

TeX:

f''(z) = \left(\frac{2 \eta}{z} + \frac{\ell \left(\ell + 1\right)}{{z}^{2}} - 1\right) f(z)\; \text{ where } f(z) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)

{c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C} \;\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(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Pow`	${a}^{b}$	Power
`CoulombF`	$F_{\ell,\eta}\!\left(z\right)$	Regular Coulomb wave function
`CoulombG`	$G_{\ell,\eta}\!\left(z\right)$	Irregular Coulomb wave function
`CC`	$\mathbb{C}$	Complex numbers
`ConstI`	$i$	Imaginary unit
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("07a654"),
    Formula(Where(Equal(ComplexDerivative(f(z), For(z, z, 2)), Mul(Sub(Add(Div(Mul(2, eta), z), Div(Mul(ell, Add(ell, 1)), Pow(z, 2))), 1), f(z))), Equal(f(z), Add(Mul(Subscript(c, 1), CoulombF(ell, eta, z)), Mul(Subscript(c, 2), CoulombG(ell, eta, z)))))),
    Variables(ell, eta, Subscript(c, 1), Subscript(c, 2), z),
    Assumptions(And(Element(Subscript(c, 1), CC), Element(Subscript(c, 2), CC), 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, OpenClosedInterval(Neg(Infinity), 0))))))

faa118

f'''(z) = \left(\frac{2 \eta}{z} + \frac{\ell \left(\ell + 1\right)}{{z}^{2}} - 1\right) f'(z) - 2 \left(\frac{\eta}{{z}^{2}} + \frac{\ell \left(\ell + 1\right)}{{z}^{3}}\right) f(z)\; \text{ where } f(z) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)

Assumptions:

{c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C} \;\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(-\infty, 0\right]

TeX:

f'''(z) = \left(\frac{2 \eta}{z} + \frac{\ell \left(\ell + 1\right)}{{z}^{2}} - 1\right) f'(z) - 2 \left(\frac{\eta}{{z}^{2}} + \frac{\ell \left(\ell + 1\right)}{{z}^{3}}\right) f(z)\; \text{ where } f(z) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)

{c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C} \;\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(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Pow`	${a}^{b}$	Power
`CoulombF`	$F_{\ell,\eta}\!\left(z\right)$	Regular Coulomb wave function
`CoulombG`	$G_{\ell,\eta}\!\left(z\right)$	Irregular Coulomb wave function
`CC`	$\mathbb{C}$	Complex numbers
`ConstI`	$i$	Imaginary unit
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("faa118"),
    Formula(Where(Equal(ComplexDerivative(f(z), For(z, z, 3)), Sub(Mul(Sub(Add(Div(Mul(2, eta), z), Div(Mul(ell, Add(ell, 1)), Pow(z, 2))), 1), ComplexDerivative(f(z), For(z, z, 1))), Mul(Mul(2, Add(Div(eta, Pow(z, 2)), Div(Mul(ell, Add(ell, 1)), Pow(z, 3)))), f(z)))), Equal(f(z), Add(Mul(Subscript(c, 1), CoulombF(ell, eta, z)), Mul(Subscript(c, 2), CoulombG(ell, eta, z)))))),
    Variables(ell, eta, Subscript(c, 1), Subscript(c, 2), z),
    Assumptions(And(Element(Subscript(c, 1), CC), Element(Subscript(c, 2), CC), 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, OpenClosedInterval(Neg(Infinity), 0))))))

eca10b

\frac{{f}^{(r + 4)}(z)}{\left(r + 4\right)!} = -\frac{1}{{z}^{2} \left({r}^{2} + 7 r + 12\right)} \left(2 \left({r}^{2} + 5 r + 6\right) z \frac{{f}^{(r + 3)}(z)}{\left(r + 3\right)!} + \left({r}^{2} + 3 r + {z}^{2} - 2 z \eta - \ell \left(\ell + 1\right) + 2\right) \frac{{f}^{(r + 2)}(z)}{\left(r + 2\right)!} + 2 \left(z - \eta\right) \frac{{f}^{(r + 1)}(z)}{\left(r + 1\right)!} + \frac{{f}^{(r)}(z)}{r !}\right)\; \text{ where } f(z) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)

Assumptions:

r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; {c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C} \;\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(-\infty, 0\right]

TeX:

\frac{{f}^{(r + 4)}(z)}{\left(r + 4\right)!} = -\frac{1}{{z}^{2} \left({r}^{2} + 7 r + 12\right)} \left(2 \left({r}^{2} + 5 r + 6\right) z \frac{{f}^{(r + 3)}(z)}{\left(r + 3\right)!} + \left({r}^{2} + 3 r + {z}^{2} - 2 z \eta - \ell \left(\ell + 1\right) + 2\right) \frac{{f}^{(r + 2)}(z)}{\left(r + 2\right)!} + 2 \left(z - \eta\right) \frac{{f}^{(r + 1)}(z)}{\left(r + 1\right)!} + \frac{{f}^{(r)}(z)}{r !}\right)\; \text{ where } f(z) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)

r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; {c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C} \;\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(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`CoulombF`	$F_{\ell,\eta}\!\left(z\right)$	Regular Coulomb wave function
`CoulombG`	$G_{\ell,\eta}\!\left(z\right)$	Irregular Coulomb wave function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`ConstI`	$i$	Imaginary unit
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("eca10b"),
    Formula(Where(Equal(Div(ComplexDerivative(f(z), For(z, z, Add(r, 4))), Factorial(Add(r, 4))), Mul(Div(-1, Mul(Pow(z, 2), Add(Add(Pow(r, 2), Mul(7, r)), 12))), Add(Add(Add(Mul(Mul(Mul(2, Add(Add(Pow(r, 2), Mul(5, r)), 6)), z), Div(ComplexDerivative(f(z), For(z, z, Add(r, 3))), Factorial(Add(r, 3)))), Mul(Add(Sub(Sub(Add(Add(Pow(r, 2), Mul(3, r)), Pow(z, 2)), Mul(Mul(2, z), eta)), Mul(ell, Add(ell, 1))), 2), Div(ComplexDerivative(f(z), For(z, z, Add(r, 2))), Factorial(Add(r, 2))))), Mul(Mul(2, Sub(z, eta)), Div(ComplexDerivative(f(z), For(z, z, Add(r, 1))), Factorial(Add(r, 1))))), Div(ComplexDerivative(f(z), For(z, z, r)), Factorial(r))))), Equal(f(z), Add(Mul(Subscript(c, 1), CoulombF(ell, eta, z)), Mul(Subscript(c, 2), CoulombG(ell, eta, z)))))),
    Variables(r, ell, eta, Subscript(c, 1), Subscript(c, 2), z),
    Assumptions(And(Element(r, ZZGreaterEqual(0)), Element(Subscript(c, 1), CC), Element(Subscript(c, 2), CC), 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, OpenClosedInterval(Neg(Infinity), 0))))))

d280c5

F_{\ell,\eta}\!\left(z\right) = C_{\ell}\!\left(\eta\right) {z}^{\ell + 1} {e}^{\omega i z} \,{}_1F_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}}\; 2 \ell + 2 \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

F_{\ell,\eta}\!\left(z\right) = C_{\ell}\!\left(\eta\right) {z}^{\ell + 1} {e}^{\omega i z} \,{}_1F_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}}\; 2 \ell + 2 \notin \{0, -1, \ldots\} \;\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
`CoulombC`	$C_{\ell}\!\left(\eta\right)$	Coulomb wave function Gamow factor
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`Hypergeometric1F1`	$\,{}_1F_1\!\left(a, b, z\right)$	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("d280c5"),
    Formula(Equal(CoulombF(ell, eta, z), Mul(Mul(Mul(CoulombC(ell, eta), Pow(z, Add(ell, 1))), Exp(Mul(Mul(omega, ConstI), z))), Hypergeometric1F1(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))), NotElement(Add(Mul(2, ell), 2), ZZLessEqual(0)), Element(z, SetMinus(CC, Set(0))))))

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))))))

1976e1

F_{\ell,\eta}\!\left(z\right) = {2}^{\ell} {z}^{\ell + 1} \exp\!\left(\frac{\log \Gamma(u) + \log \Gamma(v) - \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(v)} + \frac{{e}^{-i z} U^{*}\!\left(v, 2 \ell + 2, 2 i z\right)}{{\left(-2 i z\right)}^{v} \Gamma(u)}\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(u) + \log \Gamma(v) - \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(v)} + \frac{{e}^{-i z} U^{*}\!\left(v, 2 \ell + 2, 2 i z\right)}{{\left(-2 i z\right)}^{v} \Gamma(u)}\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`	$\log \Gamma(z)$	Logarithmic gamma function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`Gamma`	$\Gamma(z)$	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(Pi, 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), Gamma(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), Gamma(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))))))

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))))

8027e8

G_{\ell,\eta}\!\left(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) - i F_{\ell,\eta}\!\left(z\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}}\; \left(\operatorname{Im}(z) \ge 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) > 0\right)

TeX:

G_{\ell,\eta}\!\left(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) - i F_{\ell,\eta}\!\left(z\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}}\; \left(\operatorname{Im}(z) \ge 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) > 0\right)

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
`CoulombF`	$F_{\ell,\eta}\!\left(z\right)$	Regular Coulomb wave function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("8027e8"),
    Formula(Equal(CoulombG(ell, eta, z), Sub(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(ConstI, CoulombF(ell, eta, 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))), Or(GreaterEqual(Im(z), 0), Greater(Re(z), 0)))))

69e5fb

G_{\ell,\eta}\!\left(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) + i F_{\ell,\eta}\!\left(z\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}}\; \left(\operatorname{Im}(z) < 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) \ge 0\right)

TeX:

G_{\ell,\eta}\!\left(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) + i F_{\ell,\eta}\!\left(z\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}}\; \left(\operatorname{Im}(z) < 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) \ge 0\right)

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
`CoulombF`	$F_{\ell,\eta}\!\left(z\right)$	Regular Coulomb wave function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("69e5fb"),
    Formula(Equal(CoulombG(ell, eta, z), Add(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))), Mul(ConstI, CoulombF(ell, eta, 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))), Or(Less(Im(z), 0), GreaterEqual(Re(z), 0)))))

bcdfc6

H^{+}_{\ell,\eta}\!\left(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)

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}}\; \left(\operatorname{Im}(z) \ge 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) > 0\right)

TeX:

H^{+}_{\ell,\eta}\!\left(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)

\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}}\; \left(\operatorname{Im}(z) \ge 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) > 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`CoulombH`	$H^{\omega}_{\ell,\eta}\!\left(z\right)$	Outgoing and ingoing 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
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("bcdfc6"),
    Formula(Equal(CoulombH(1, ell, eta, z), 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)))))),
    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))), Or(GreaterEqual(Im(z), 0), Greater(Re(z), 0)))))

f0414a

H^{+}_{\ell,\eta}\!\left(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) + 2 i F_{\ell,\eta}\!\left(z\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}}\; \left(\operatorname{Im}(z) < 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) \ge 0\right)

TeX:

H^{+}_{\ell,\eta}\!\left(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) + 2 i F_{\ell,\eta}\!\left(z\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}}\; \left(\operatorname{Im}(z) < 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) \ge 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`CoulombH`	$H^{\omega}_{\ell,\eta}\!\left(z\right)$	Outgoing and ingoing 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
`CoulombF`	$F_{\ell,\eta}\!\left(z\right)$	Regular Coulomb wave function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("f0414a"),
    Formula(Equal(CoulombH(1, ell, eta, z), Add(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))), Mul(Mul(2, ConstI), CoulombF(ell, eta, 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))), Or(Less(Im(z), 0), GreaterEqual(Re(z), 0)))))

781eae

H^{-}_{\ell,\eta}\!\left(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)

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}}\; \left(\operatorname{Im}(z) < 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) \ge 0\right)

TeX:

H^{-}_{\ell,\eta}\!\left(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)

\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}}\; \left(\operatorname{Im}(z) < 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) \ge 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`CoulombH`	$H^{\omega}_{\ell,\eta}\!\left(z\right)$	Outgoing and ingoing 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
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("781eae"),
    Formula(Equal(CoulombH(-1, ell, eta, 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))), Or(Less(Im(z), 0), GreaterEqual(Re(z), 0)))))

0cc301

H^{-}_{\ell,\eta}\!\left(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) - 2 i F_{\ell,\eta}\!\left(z\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}}\; \left(\operatorname{Im}(z) \ge 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) > 0\right)

TeX:

H^{-}_{\ell,\eta}\!\left(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) - 2 i F_{\ell,\eta}\!\left(z\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}}\; \left(\operatorname{Im}(z) \ge 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) > 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`CoulombH`	$H^{\omega}_{\ell,\eta}\!\left(z\right)$	Outgoing and ingoing 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
`CoulombF`	$F_{\ell,\eta}\!\left(z\right)$	Regular Coulomb wave function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("0cc301"),
    Formula(Equal(CoulombH(-1, ell, eta, z), Sub(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(2, ConstI), CoulombF(ell, eta, 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))), Or(GreaterEqual(Im(z), 0), Greater(Re(z), 0)))))

Definitions

Differential equations

Connection formulas

Normalization functions

Derivatives

Hypergeometric representations

Kummer function

Tricomi function