Beam-beam amplitude detuning with forced oscillations

Recently, relations were established between the coefficients of free and forced amplitude detuning polynomial expansions. The forced oscillations were considered only in a single plane. In this paper we extend and generalize previous results by developing analytical equations that transform the free amplitude detuning function into the amplitude detuning involving forced oscillations in both transverse planes. These are used to obtain closed approximated formulas for the beam-beam amplitude detuning with forced oscillations. Formulas are compared to single and multiparticle simulations.