  It is typical in the accelerator field to model machine components,
especially RF cavities, as parallel RLC resonators. To properly model
wake-fields, knowledge of the time-domain voltage resulting from beam
excitation is often necessary. While analytical and quasi-analytical
expressions are available to accomplish this for common bunch distributions
such as the Gaussian, analogous results for less standard distributions can be
difficult or computationally-taxing to obtain using direct methods, which opens
the door for the development of a more generalized technique. In this paper, a
formulation is created that allows for the simple computation of the
time-domain voltage waveform of an RLC resonator. The formulation uses the
Cauchy Residue Theorem to extract the convolution result from the Fourier
Domain, and if current distribution Fourier Transform has no poles, knowledge
of its value is only required at one specific evaluation point. This greatly
simplifies the computation of the time domain voltage for a large amount of
bunch distributions both common and uncommon. Accuracy considerations for this
technique and the approximation of accelerator components as RLC resonators are
also discussed, resulting the development of a figure of merit for quantifying
the robustness of this type of approximation.
