by our method agree with the established approximation C/Re[Y] away from resonance and do not diverge near resonance. Additionally, we demonstrate that a system of two resonators coupled to a common Purcell filter exhibits a pattern of hybridization that cannot be explained by a hierarchy of couplings and detunings. This is due to the significant influence of radiative loss on the mode couplings. As a result, it is necessary to include radiative boundary conditions even in simulations of coherent dynamics. Our simulation technique is useful for designing complex circuit quantum electrodynamic systems.