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.
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Footnotes
Based on H. E. Salzer, Math. Tables Aids Comp. (1955), 9, 164; J. Maths. Phys. (1958), 37, 89.
For example, see STROUD (1974), pp. 135ff.
H. E. Salzer, J. Maths. Phys. (1961), 40, 72; STROUD $ SECREST (1966), pp. 307ff.
This argument is given in LUKE (1969), vol. II, pp. 253ff.
We have corrected Piessens’ formulas for the coefficients to remove some errors.
RIVLIN (1974), p. 47.
Based on A. Papoulis, Q. Appl. Math. (1956), 14, 405.
ABRAMOWITZ $ STEGUN (1965), Ch. 22.
A very thorough treatment may be found in LUKE (1969), vol. II, Ch. 10.
I. M. Longman, Int. J. Comp. Math. B (1971), 3, 53.
Obviously such a curcumstance would cause peculiar difficulties.
Some other possibilities for the use of Padé approximation are discussed in LUKE (1969), vol. II, pp. 255ff.
See I. M. Longman, M. Sharir, Geophys. J. Roy. Astr. Soc. (1971), 25, 299.
I. M. Longman, J. Comp. Phys. (1972), 10, 224.
This method is the subject of BELLMAN, KALABA E LOCKETT (1966).
LUKE (1969), vol. II, pp. 247–251.
W. T. Weeks, J. Ass. Comp. Mach. (1966), 13, 419;
R. A. Spinelli, SIAM J. Num. Anal. (1966), 3, 636.
See also R. Piessens & M. Branders, Proc. I.R.E.E. (1971), 118, 1517 for a generalization.
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Davies, B. (1978). Numerical Inversion of Laplace Transforms. In: Integral Transforms and Their Applications. Applied Mathematical Sciences, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-5512-1_21
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DOI: https://doi.org/10.1007/978-1-4757-5512-1_21
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