Convolution of a time-dependent driven excitation into the decay curve of an arbitrarily cascaded and blended level

An analysis is presented of the instantaneous emission rate from an atomic level that is populated by arbitrary conditions of both collisional excitation and cascade repopulation. The general solution to the rate equations is obtained in terms of transition probabilities, excitation cross sections, and a simple integral operator, and applications are given for the specific examples of impulsive, stepwise, rectangular, gaussian, and modulated excitation conditions.