Abstract
There are many problems whose solution may be found in terms of a Laplace or Fourier transform, which is 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
A more comprehensive survey and evaluation may be found in B. Davies and B. Martin, J. Comp. Phys. (1979), 33
Based on H. E. Salzer, Math. Tables and other aids to computation, (1955), 9, 164; Journal of Maths. and Phys. (1958), 37, 89.
For example, see Stroud (1974).
H. E. Salzer, Journal of Math. and Phys. (1961), 40, 72: Stroud & Secrest (1966).
This argument is given in Luke (1969), vol. II.
Laguerre polynomials were suggested by F. Tricomi, R. C. Acad. Nat. dei Lincei 21 (1935), 232 and D. V. Widder, Duke Math. J., 1 (1935), 126. Their practical use was developed by W. T. Weeks, J. ACM. 13 (1966), 419 and R. Piessens and M. Branders, Proc. IEEE, 118 (1971), 1
W. T. Weeks, J. ACM, 13 (1966), 419.
Based on R. Piessens, J. Inst. Maths. Applics. (1972), 10, 185. In the original paper, Piessens writes where theere are Jacobi polynomials. We consider only the special case α = β = −1/2, which forms the main body of Piessens’ pap
We have corrected Piessens formulae for the coefficients to remove some errors.
Rivlin (1974), p. 47.
H. Dubner and J. Abate, J. ACM. 15 (1968), 115. M. Silverberg, Electron Lett. 6 (1970), 105.
F. Durbin, Comput. J., 17 (1974), 371.
J. R. MacDonald, J. Appl. Phys. 35 (1964); 3034.
J. W. Cooley and J. W. Tukey, Math. Comp., 19 (1965), 297.
A very thorough treatment may be found in Luke (1969), vol. II.
I. M. Longman, Int. J. Comp. Math. B, (1971), 3, 53.
Obviously such a circumstance would cause peculiar difficulties.
Some other possibilities for the use of Pade approximation are discussed in Luke (1969), vol. II.
See I. M. Longman & M. Sharir, Geophys. J. Roy. Astr. Soc, (1971), 25, 299.
I. M. Longman, J. Comp. Phys. (1972), 10, 224.
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Davies, B. (1985). Numerical Inversion of Laplace Transforms. In: Integral Transforms and their Applications. Applied Mathematical Sciences, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4899-2691-3_21
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DOI: https://doi.org/10.1007/978-1-4899-2691-3_21
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