# Coulomb wave functions

## Definitions

Symbol: CoulombF $F_{\ell,\eta}\!\left(z\right)$ Regular Coulomb wave function
Symbol: CoulombG $G_{\ell,\eta}\!\left(z\right)$ Irregular Coulomb wave function
Symbol: CoulombH $H^{\omega}_{\ell,\eta}\!\left(z\right)$ Outgoing and ingoing Coulomb wave function
Symbol: CoulombC $C_{\ell}\!\left(\eta\right)$ Coulomb wave function Gamow factor
Symbol: CoulombSigma $\sigma_{\ell}\!\left(\eta\right)$ Coulomb wave function phase shift

## Differential equations

$y''(z) + \left(1 - \frac{2 \eta}{z} - \frac{\ell \left(\ell + 1\right)}{{z}^{2}}\right) y(z) = 0\; \text{ where } y(z) = {c}_{1} F_{\ell,\eta}\!\left(z\right) + {c}_{2} G_{\ell,\eta}\!\left(z\right)$
$G_{\ell,\eta}\!\left(z\right) \left(\frac{d}{d z}\, F_{\ell,\eta}\!\left(z\right)\right) - \left(\frac{d}{d z}\, G_{\ell,\eta}\!\left(z\right)\right) F_{\ell,\eta}\!\left(z\right) = 1$

## Connection formulas

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

## Normalization functions

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

## Derivatives

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

## Hypergeometric representations

### Kummer function

$F_{\ell,\eta}\!\left(z\right) = C_{\ell}\!\left(\eta\right) {z}^{\ell + 1} {e}^{\omega i z} \,{}_1F_1\!\left(1 + \ell + \omega i \eta, 2 \ell + 2, -2 \omega i z\right)$
$F_{\ell,\eta}\!\left(z\right) = {2}^{\ell} \exp\!\left(\frac{\log \Gamma\!\left(1 + \ell + i \eta\right) + \log \Gamma\!\left(1 + \ell - i \eta\right) - \pi \eta}{2}\right) {z}^{\ell + 1} {e}^{\omega i z} \,{}_1{\textbf F}_1\!\left(1 + \ell + \omega i \eta, 2 \ell + 2, -2 \omega i z\right)$

### Tricomi function

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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC