It is known that amplitudes which differ from the Coulomb one by an overall phase factor and by a distribution with a support at zero scattering angle, describe the same scattering process. We utilize this fact to derive new partial-wave expansions, which have finite expansion coefficients, for amplitudes of Coulomb-like interactions. A modified form of the Lippmann-Schwinger equation is derived. For the case of the Coulomb interaction this equation leads to a different amplitude from the Coulomb one, but equivalent to it as both describe the same scattering process. The method can be extended to derive (free of infinities) partial-wave expansions of some field theoretical amplitudes.