Large-time Behaviour of Wave Packets

  • Antonio BarlettaEmail author


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.


  1. 1.
    Ablowitz MJ, Fokas AS (2003) Complex variables: introduction and applications. Cambridge University Press, CambridgeGoogle Scholar
  2. 2.
    Abramowitz M, Stegun I (1968) Handbook of mathematical functions. Dover, New YorkGoogle Scholar
  3. 3.
    Apostol TM (1967) Calculus, vol. 2: multi-variable calculus and linear algebra with applications to differential equations and probability. Wiley, New YorkGoogle Scholar
  4. 4.
    Arfken GB, Weber HJ, Harris FE (2012) Mathematical methods for physicists: a comprehensive guide. Elsevier, New YorkzbMATHGoogle Scholar
  5. 5.
    Bender CM, Orszag SA (1999) Advanced mathematical methods for scientists and engineers I. Springer, New YorkCrossRefGoogle Scholar
  6. 6.
    Cartan H (1995) Elementary theory of analytic functions of one or several complex variables. Dover, New YorkGoogle Scholar
  7. 7.
    Debnath L, Bhatta D (2014) Integral transforms and their applications. CRC Press, New YorkGoogle Scholar
  8. 8.
    Priestley HA (2003) Introduction to complex analysis. Oxford University Press, OxfordGoogle Scholar
  9. 9.
    Schiff JL (1999) The Laplace transform: theory and applications. Springer, New YorkCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Industrial EngineeringAlma Mater Studiorum Università di BolognaBolognaItaly

Personalised recommendations