Do you mean the Born approximation? That's just the 1st-order scattering amplitude in usual perturbation theory. You find it in any good textbook on quantum theory. For the formalism, I'd recommend Sakurai, Modern Quantum Mechanics, and to understand the physics and the assumptions underlying the definition of the S-matrix, read the corresponding chapter in Mesiah, Quantum Mechanics, which gives a very concise analysis in terms of the wave-packet approach.