Coulomb wave function

Remarks

The 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 $\rho \to 0^+$. The 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 $\theta_L = \rho - \eta \log 2\rho - \frac{L\pi}{2} + \arg \Gamma(L + 1 + i\eta)$. The irregular Coulomb wave function $G_L(\eta, \rho)$ is the solution with $G_L(\eta, \rho) = \cos \theta_L(\eta, \rho) + o(1)$ as $\rho \to \infty$. Example: 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 $$F_0(0, \rho) = \rho\, e^{-i\rho} \frac{e^{2i\rho} - 1}{2i\rho} = \frac{e^{i\rho} - e^{-i\rho}}{2i} = \sin \rho.$$ Non-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 $G_0(0, \rho)$. For $\eta = 0$, $F_L(0, \rho) = \rho\, j_L(\rho)$, where $j_L$ is the spherical Bessel function of the first kind.