Large-time Behaviour of Wave Packets

This chapter aims to illustrate the saddle-point approximation as a tool to detect the behaviour at large times of wave packets. This objective can be achieved by displaying techniques based on the theory of holomorphic functions of a complex variable. The reader is guided along this way by a brief illustration of the main features of complex variables and holomorphic functions. Properties of integrals over paths in the complex plane are discussed. Elements of Laurent series expansions, singular points, residues and a statement of Cauchy’s residue theorem are provided. The main features of the Laplace transform are surveyed. Then, the behaviour of a wave packet in the asymptotic regime of large time is studied and the saddle-point approximation is presented. The central role of homotopy, that is the possibility of deforming continuously a path in the complex plane, is discussed.