In nuclear physics, the Bateman equation is a mathematical model describing abundances and activities in a decay chain as a function of time, based on the decay rates and initial abundances. If, at time t, there are atoms of isotope that decays into isotope at the rate , the amounts of isotopes in the k-step decay chain evolves as: (this can be adapted to handle decay branches). While this can be solved explicitly for i=2, the formulas quickly become cumbersome for longer chains. Bateman found a general explicit formula for the amounts by taking the Laplace transform of the variables.