Asymptotic Methods of Analysis using Advanced Saddle Point Techniques

  • Kurt E. Oughstun
Part of the Springer Series in Optical Sciences book series (SSOS, volume 144)


The integral representation developed in Vol. 1 and reviewed in Chap. 9 of this volume provides an exact, formal solution to the problem of electromagnetic pulse propagation in homogeneous, isotropic, locally linear, temporally dispersive media either filling all of space or filling the half-space z > z 0. However, an exact, analytic evaluation of the resultant contour integral is typically not possible for a rather broad class of realistic initial pulse shapes. Consequently, a well-defined approximate evaluation of the integral representation for a given initial pulse is necessary in order to determine the behavior of the temporal phenomena of primary interest here. This includes the spatiotemporal properties of the precursor fields, the arrival of the signal, the signal velocity, and the spatiotemporal evolution of the pulse. To have complete confidence in the results, it is essential that this approxiamate evaluation procedure possess a useful, well-defined error bound.


Saddle Point Asymptotic Expansion Steep Descent Asymptotic Approximation Airy Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914.Google Scholar
  2. 2.
    L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960.zbMATHGoogle Scholar
  3. 3.
    F. W. J. Olver, “Why steepest descents?,” SIAM Rev., vol. 12, no. 2, pp. 228–247, 1970.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978.Google Scholar
  5. 5.
    K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B, vol. 5, no. 4, pp. 817–849, 1988.ADSCrossRefGoogle Scholar
  6. 6.
    R. A. Handelsman and N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Rat. Mech. Anal., vol. 35, pp. 267–283, 1969.MathSciNetzbMATHGoogle Scholar
  7. 7.
    K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A, vol. 6, no. 9, pp. 1394–1420, 1989.MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Phil. Soc., vol. 53, pp. 599–611, 1957.MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1973.Google Scholar
  10. 10.
    J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, and P. Smith, “Uniform asymptotic description of the Brillouin precursor in a single-resonance Lorentz model dielectric,” Pure Appl. Opt., vol. 7, no. 3, pp. 575–602, 1998.ADSCrossRefGoogle Scholar
  11. 11.
    N. Bleistein, “Uniform asymptotic expansions of integrals with stationary point near algebraic singularity,” Com. Pure Appl. Math., vol. XIX, no. 4, pp. 353–370, 1966.MathSciNetCrossRefGoogle Scholar
  12. 12.
    N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,” J. Math. Mech, vol. 17, no. 6, pp. 533–559, 1967.MathSciNetzbMATHGoogle Scholar
  13. 13.
    N. A. Cartwright, Uniform Asymptotic Description of the Unit Step Function Modulated Sinusoidal Signal. PhD thesis, College of Engineering & Mathematical Sciences, University of Vermont, 2004.Google Scholar
  14. 14.
    N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev., vol. 49, no. 4, pp. 628–648, 2007.MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    W. Fulks and J. O. Sather, “Asymptotics II: Laplace’s method for multiple integrals,” Pacific J. of Math., vol. 11, pp. 185–192, 1961.MathSciNetzbMATHGoogle Scholar
  16. 16.
    B. Riemann, Gesammelte Mathematische Werke. Leipzig: Teubner, 1876.zbMATHGoogle Scholar
  17. 17.
    P. Debye, “Näherungsformeln für die zylinderfunktionen für grosse werte des arguments und unbeschränkt verander liche werte des index,” Math Ann., vol. 67, pp. 535–558, 1909.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys., vol. 44, pp. 177–202, 1914.CrossRefGoogle Scholar
  19. 19.
    K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: Springer-Verlag, 1994.Google Scholar
  20. 20.
    P. M. Morse and H. Feshbach, Methods of Theoretical Physics. New York: McGraw-Hill, 1953. Vol. I.Google Scholar
  21. 21.
    E. T. Copson, Asymptotic Expansions. London: Cambridge University Press, 1965.zbMATHCrossRefGoogle Scholar
  22. 22.
    M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, vol. 55 of Applied Mathematics Series. Washington, D.C.: National Bureau of Standards, 1964.Google Scholar
  23. 23.
    H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering. Cambridge: Cambridge University Press, 1992.CrossRefGoogle Scholar
  24. 24.
    J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves. Bristol, England: Adam Hilger, 1986.zbMATHGoogle Scholar
  25. 25.
    E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. p. 110.Google Scholar
  26. 26.
    L. B. Felsen and N. Marcuvitz, “Modal analysis and synthesis of electromagnetic fields,” Polytechnic Inst. Brooklyn, Microwave Res. Inst. Rep., 1959.Google Scholar
  27. 27.
    A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space. Oxford: Pergamon Press, 1966. Sect. 3.3.Google Scholar
  28. 28.
    L. C. Hsu, “On the asymptotic evaluation of a class of multiple integrals involving a parameter,” Duke Math. J., vol. 15, pp. 625–634, 1948.CrossRefGoogle Scholar
  29. 29.
    N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals. New York: Holt, Rinehart and Winston, 1975.Google Scholar
  30. 30.
    F. W. J. Olver, Asymptotics and Special Functions. Natick: A K Peters, 1997.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Kurt E. Oughstun
    • 1
  1. 1.College of Engineering & Mathematical Sciences School of EngineeringUniversity of VermontBurlingtonUSA

Personalised recommendations