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Coulomb wave functions

Table of contents: Definitions - Differential equations - Connection formulas - Normalization functions - Derivatives - Hypergeometric representations

Definitions

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

Differential equations

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y(z)+(12ηz(+1)z2)y(z)=0   where y(z)=c1F,η ⁣(z)+c2G,η ⁣(z)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)
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G,η ⁣(z)(ddzF,η ⁣(z))(ddzG,η ⁣(z))F,η ⁣(z)=1G_{\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

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F,η ⁣(z)=H,η+ ⁣(z)H,η ⁣(z)2iF_{\ell,\eta}\!\left(z\right) = \frac{H^{+}_{\ell,\eta}\!\left(z\right) - H^{-}_{\ell,\eta}\!\left(z\right)}{2 i}
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G,η ⁣(z)=H,η+ ⁣(z)+H,η ⁣(z)2G_{\ell,\eta}\!\left(z\right) = \frac{H^{+}_{\ell,\eta}\!\left(z\right) + H^{-}_{\ell,\eta}\!\left(z\right)}{2}
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H,ηω ⁣(z)=G,η ⁣(z)+ωiF,η ⁣(z)H^{\omega}_{\ell,\eta}\!\left(z\right) = G_{\ell,\eta}\!\left(z\right) + \omega i F_{\ell,\eta}\!\left(z\right)
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G,η ⁣(z)=F,η ⁣(z)cos(χ)F1,η ⁣(z)sin(χ)   where χ=σ ⁣(η)σ1 ⁣(η)(+12)πG_{\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
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H,ηω ⁣(z)=F,η ⁣(z)eωiχF1,η ⁣(z)sin(χ)   where χ=σ ⁣(η)σ1 ⁣(η)(+12)πH^{\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

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C ⁣(η)=2Γ ⁣(2+2)exp ⁣(logΓ ⁣(1++iη)+logΓ ⁣(1+iη)πη2)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)
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σ ⁣(η)=logΓ ⁣(1++iη)logΓ ⁣(1+iη)2i\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

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ddzF,η ⁣(z)=(+1z+η+1)F,η ⁣(z)1++iη1+iη+1F+1,η ⁣(z)\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)
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ddzG,η ⁣(z)=(+1z+η+1)G,η ⁣(z)1++iη1+iη+1G+1,η ⁣(z)\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)
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f(z)=(2ηz+(+1)z21)f(z)   where f(z)=c1F,η ⁣(z)+c2G,η ⁣(z)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)
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f(z)=(2ηz+(+1)z21)f(z)2(ηz2+(+1)z3)f(z)   where f(z)=c1F,η ⁣(z)+c2G,η ⁣(z)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)
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f(r+4)(z)(r+4)!=1z2(r2+7r+12)(2(r2+5r+6)zf(r+3)(z)(r+3)!+(r2+3r+z22zη(+1)+2)f(r+2)(z)(r+2)!+2(zη)f(r+1)(z)(r+1)!+f(r)(z)r!)   where f(z)=c1F,η ⁣(z)+c2G,η ⁣(z)\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

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F,η ⁣(z)=C ⁣(η)z+1eωiz1F1 ⁣(1++ωiη,2+2,2ωiz)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)
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F,η ⁣(z)=2exp ⁣(logΓ ⁣(1++iη)+logΓ ⁣(1+iη)πη2)z+1eωiz1F1 ⁣(1++ωiη,2+2,2ωiz)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

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F,η ⁣(z)=2z+1exp ⁣(logΓ(u)+logΓ(v)πη2)(eizU ⁣(u,2+2,2iz)(2iz)uΓ(v)+eizU ⁣(v,2+2,2iz)(2iz)vΓ(u))   where u=1++iη,v=1+iηF_{\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
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Gl,η ⁣(z)=12((2z)iηexp ⁣(i(zπ2+σ ⁣(η)))U ⁣(1++iη,2+2,2iz)+(2z)iηexp ⁣(i(zπ2+σ ⁣(η)))U ⁣(1+iη,2+2,2iz))G_{l,\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)
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Gl,η ⁣(z)=(2z)iηexp ⁣(i(zπ2+σ ⁣(η)))U ⁣(1++iη,2+2,2iz)iF,η ⁣(z)G_{l,\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)
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Gl,η ⁣(z)=(2z)iηexp ⁣(i(zπ2+σ ⁣(η)))U ⁣(1+iη,2+2,2iz)+iF,η ⁣(z)G_{l,\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)
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Hl,η+ ⁣(z)=(2z)iηexp ⁣(i(zπ2+σ ⁣(η)))U ⁣(1++iη,2+2,2iz)H^{+}_{l,\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)
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Hl,η+ ⁣(z)=(2z)iηexp ⁣(i(zπ2+σ ⁣(η)))U ⁣(1+iη,2+2,2iz)+2iF,η ⁣(z)H^{+}_{l,\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)
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Hl,η ⁣(z)=(2z)iηexp ⁣(i(zπ2+σ ⁣(η)))U ⁣(1+iη,2+2,2iz)H^{-}_{l,\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)
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Hl,η ⁣(z)=(2z)iηexp ⁣(i(zπ2+σ ⁣(η)))U ⁣(1++iη,2+2,2iz)2iF,η ⁣(z)H^{-}_{l,\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.

2019-11-11 15:50:15.016492 UTC