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  "body": "\n## Definition\n\nLet $L$ be a nonnegative integer, let $\\eta$ be a real number, and let $\\rho \u003e 0$.\nThe *Coulomb wave equation* is the [[linear differential equation|0QQRP17Z7S]]\n$$\\frac{d^2 u}{d\\rho^2} + \\left( 1 - \\frac{2\\eta}{\\rho} - \\frac{L(L + 1)}{\\rho^2} \\right) u = 0.$$\nLet $M(a, b, z)$ be [[Kummer's function|6AZ4ZBN1M8]] and $\\Gamma$ the [[gamma function|0H5SYBSED9]].\nThe *regular Coulomb wave function* is\n$$F_L(\\eta, \\rho) = C_L(\\eta)\\, \\rho^{L + 1} e^{-i\\rho}\\, M(L + 1 - i\\eta,\\ 2L + 2,\\ 2i\\rho),$$\nwhere the *Coulomb normalising constant* is\n$$C_L(\\eta) = \\frac{2^L e^{-\\pi\\eta/2} \\abs{\\Gamma(L + 1 + i\\eta)}}{\\Gamma(2L + 2)}.$$\n\n## Remarks\n\nThe function $F_L(\\eta, \\rho)$ is real, solves the Coulomb wave equation, and satisfies $F_L(\\eta, \\rho) \\sim C_L(\\eta) \\rho^{L + 1}$ as $\\rho \\to 0^+$.\nThe constant $C_L(\\eta)$ is chosen so that $F_L(\\eta, \\rho) = \\sin \\theta_L(\\eta, \\rho) + o(1)$ as $\\rho \\to \\infty$, with phase $\\theta_L = \\rho - \\eta \\log 2\\rho - \\frac{L\\pi}{2} + \\arg \\Gamma(L + 1 + i\\eta)$.\nThe *irregular Coulomb wave function* $G_L(\\eta, \\rho)$ is the solution with $G_L(\\eta, \\rho) = \\cos \\theta_L(\\eta, \\rho) + o(1)$ as $\\rho \\to \\infty$.\nExample: for $\\eta = 0$ and $L = 0$, one has $C_0(0) = 1$ and $M(1, 2, 2i\\rho) = \\frac{e^{2i\\rho} - 1}{2i\\rho}$, so\n$$F_0(0, \\rho) = \\rho\\, e^{-i\\rho} \\frac{e^{2i\\rho} - 1}{2i\\rho} = \\frac{e^{i\\rho} - e^{-i\\rho}}{2i} = \\sin \\rho.$$\nNon-example: $\\cos \\rho$ also solves the equation for $\\eta = 0$ and $L = 0$, but it is not a multiple of $F_0(0, \\rho)$, since it does not vanish at $\\rho = 0$; it is $G_0(0, \\rho)$.\nFor $\\eta = 0$, $F_L(0, \\rho) = \\rho\\, j_L(\\rho)$, where $j_L$ is the spherical Bessel function of the first kind.\n",
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