Integral equations for Coulomb scattering wave functions and Coulomb asymptotic states

The connection between the homogeneous and inhomogeneous equations for the Coulomb scattering wave function of two particles is investigated. It is shown that the form of the equation depends on the method used to regularize the divergent integrals in the homogeneous part of the equation. This result is a generalization of the result obtained by Van Iiaeringen for orbital angular momentum $l=0$. It is also shown to be helpful to introduce a Coulomb asymptotic state in the momentum representation; this is the inhomogeneous part of the equation and contains all the principal information about the forward scattering of charged particles. Therefore, the Coulomb asymptotic states can be used to find the behavior of the reaction amplitudes of charged particles near singularities in $\cos\theta$, where $\theta$ is the scattering angle.