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Numerical Inversion of Laplace Transforms

  • B. Davies
Part of the Applied Mathematical Sciences book series (AMS, volume 25)

Abstract

There are many problems whose solutions may be found in terms of Laplace or Fourier transforms which are then too complicated for inversion using the techniques of complex analysis. In this section we discuss some of the methods which have been developed -- and in some cases are still being developed -- for the numerical evaluation of the Laplace inversion integral. We make no explicit reference to inverse Fourier transforms, although they may obviously be treated by similar methods because of the close relationship between the two transforms.

Keywords

Special Technique Chebyshev Polynomial Inverse Fourier Transform Numerical Inversion Laplace Inversion 
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.

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Footnotes

  1. 1.
    Based on H. E. Salzer, Math. Tables Aids Comp. (1955), 9, 164; J. Maths. Phys. (1958), 37, 89.Google Scholar
  2. 2.
    For example, see STROUD (1974), pp. 135ff.Google Scholar
  3. 3.
    H. E. Salzer, J. Maths. Phys. (1961), 40, 72; STROUD $ SECREST (1966), pp. 307ff.Google Scholar
  4. 4.
    This argument is given in LUKE (1969), vol. II, pp. 253ff.Google Scholar
  5. 6.
    We have corrected Piessens’ formulas for the coefficients to remove some errors.Google Scholar
  6. 7.
    RIVLIN (1974), p. 47.Google Scholar
  7. 8.
    Based on A. Papoulis, Q. Appl. Math. (1956), 14, 405.Google Scholar
  8. 9.
    ABRAMOWITZ $ STEGUN (1965), Ch. 22.Google Scholar
  9. 10.
    A very thorough treatment may be found in LUKE (1969), vol. II, Ch. 10.Google Scholar
  10. 11.
    I. M. Longman, Int. J. Comp. Math. B (1971), 3, 53.Google Scholar
  11. 12.
    Obviously such a curcumstance would cause peculiar difficulties.Google Scholar
  12. 13.
    Some other possibilities for the use of Padé approximation are discussed in LUKE (1969), vol. II, pp. 255ff.Google Scholar
  13. 14.
    See I. M. Longman, M. Sharir, Geophys. J. Roy. Astr. Soc. (1971), 25, 299.Google Scholar
  14. 15.
    I. M. Longman, J. Comp. Phys. (1972), 10, 224.Google Scholar
  15. 16.
    This method is the subject of BELLMAN, KALABA E LOCKETT (1966).Google Scholar
  16. 17.
    LUKE (1969), vol. II, pp. 247–251.Google Scholar
  17. 18.
    W. T. Weeks, J. Ass. Comp. Mach. (1966), 13, 419;Google Scholar
  18. R. A. Spinelli, SIAM J. Num. Anal. (1966), 3, 636.Google Scholar
  19. See also R. Piessens & M. Branders, Proc. I.R.E.E. (1971), 118, 1517 for a generalization.Google Scholar

Copyright information

© Springer Science+Business Media New York 1978

Authors and Affiliations

  • B. Davies
    • 1
  1. 1.The Australian National UniversityCanberraAustralia

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