Real Variable Inverse Laplace Transform

  • Vu Kim TuanEmail author
  • A. Boumenir
  • Dinh Thanh Duc
Part of the Trends in Mathematics book series (TM)


The aim of this work is to provide a review of authors’ contributions to the field of the Laplace transform in the last 20 years.


Inverse Laplace transform Dirichlet series Hardy space 

Mathematics Subject Classification (2010)

Primary 44A10 Secondary 30B50 30H10 65M32 



The work of the third author was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2017.310.


  1. 1.
    J. Abate, W. Whitt, The Fourier-series method for inverting transforms of probability distributions. Queueing Syst. 10, 5–88 (1992)MathSciNetCrossRefGoogle Scholar
  2. 2.
    L. Amério, Sulla transformata doppia di Laplace. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. 12, 707–780 (1941)zbMATHGoogle Scholar
  3. 3.
    R. Bellman, R.E. Kabala, R.A. Lockett, Numerical Inversion of the Laplace Transform (Elsevier, New York, 1966)Google Scholar
  4. 4.
    A. Boumenir, A. Al-Shuaibi, The inverse Laplace transform and analytic pseudo-differential operators. J. Math. Anal. Appl. 228(1), 16–36 (1998)MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Boumenir, A. Al-Shuaibi, On the numerical inversion of the Laplace transform by the use of optimized Legendre polynomials. Approx. Theory Appl. 16(4), 17–32 (2000)MathSciNetzbMATHGoogle Scholar
  6. 6.
    A. Boumenir, V.K. Tuan, Sampling eigenvalues in Hardy spaces. SIAM J. Numer. Anal. 45(2), 473–483 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Boumenir, V.K. Tuan, The computation of eigenvalues of singular Sturm-Liouville operators. Adv. Appl. Math. 39, 222–236 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Boumenir, V.K. Tuan, The interpolation of the Titchmarsh-Weyl function. J. Math. Anal. Appl. 335, 72–78 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Boumenir, V.K. Tuan, Representation and sampling of Hardy functions. Math. Methods Appl. Sci. 33(4), 485–492 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    A. Boumenir, V.K. Tuan, An inverse problem for the heat equation. Proc. Am. Math. Soc. 138(11), 3911–3921 (2010)CrossRefGoogle Scholar
  11. 11.
    A. Boumenir, V.K. Tuan, Recovery of a heat equation by four measurements at one end. Numer. Funct. Anal. Optim. 31(2), 155–163 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    L.P. Castro, H. Fujiwara, M.M. Rodrigues, S. Saitoh, A new discretization method by means of reproducing kernels, in Proceedings of the 20th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications, Hanoi University of Science and Technology, Hanoi, July 29–August 3, 2012Google Scholar
  13. 13.
    A.H.-D. Cheng, P. Sidauruk, Y. Abousleiman, Approximate inversion of the Laplace transform. Math. J. 4(2), 76–82 (1994)Google Scholar
  14. 14.
    B. Davies, B. Martin, Numerical inversion of the Laplace transform: a survey and comparison of methods. J. Comput. Phys. 33, 1–32 (1979)MathSciNetCrossRefGoogle Scholar
  15. 15.
    G. Doetsch, Handbuch der Laplace Transformation, vol. 1 (Birkhäuser, Basel, 1950)CrossRefGoogle Scholar
  16. 16.
    D.G. Duffy, On the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applications. ACM Trans. Math. Softw. 19(3), 333–359 (1993)MathSciNetCrossRefGoogle Scholar
  17. 17.
    G. Honig, U. Hirdes, Algorithm 27: a method for the numerical inversion of Laplace transforms. J. Comput. Appl. Math. 10, 113–132 (1984)MathSciNetCrossRefGoogle Scholar
  18. 18.
    V.I. Krylov, N.S. Skoblya, A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transform (Mir, Moscow, 1977)Google Scholar
  19. 19.
    B.M. Levitan, Inverse Sturm-Liouville Problems (VNU Science Press, Utrecht, 1987)CrossRefGoogle Scholar
  20. 20.
    A. Murli, M. Rizzardi, Algorithm 682: Talbot’s method for the Laplace inversion problem. AMS Trans. Math. Softw. 16(2), 158–168 (1990)CrossRefGoogle Scholar
  21. 21.
    R.E.A.C. Paley, N. Wiener, Fourier Transforms in the Complex Domain (Colloquium Publications American Mathematical Society, Providence, 1934)zbMATHGoogle Scholar
  22. 22.
    J. Peng, S.K. Chung, Laplace transforms and generators of semigroups of operators. Proc. Am. Math. Soc. 126(8), 2407–2416 (1998)MathSciNetCrossRefGoogle Scholar
  23. 23.
    R. Piessens, Algorithm 453: Gaussian quadrature formulas for Bromwich’s integral [D1]. Commun. ACM 6(8), 486–487 (1973)CrossRefGoogle Scholar
  24. 24.
    R. Piessens, R. Huysmans, Algorithm 619: automatic numerical inversion of the Laplace transform [D5]. ACM Trans. Math. Softw. 10, 348–353 (1984)CrossRefGoogle Scholar
  25. 25.
    E.L. Post, Generalized differentiation. Trans. Am. Math. Soc. 32, 723–781 (1930)MathSciNetCrossRefGoogle Scholar
  26. 26.
    A.P. Prudnikov, Y.A. Brychkov, O.I. Marichev, Integrals and Series. Volume 4: Direct Laplace Transforms (Gordon and Breach, New York, 1992)Google Scholar
  27. 27.
    A. Rybkin, V.K. Tuan, A new interpolation formula for the Titchmarsch-Weyl m-function. Proc. Am. Math. Soc. 137(12), 4177–4185 (2009)CrossRefGoogle Scholar
  28. 28.
    H. Stehfest, Algorithm 368: numerical inversion of Laplace transforms [D5]. Commun. ACM 13(1), 47–49 (1970)CrossRefGoogle Scholar
  29. 29.
    V.K. Tuan, On the factorization of integral transformations of convolution type in the space \(L_{2}^{\Phi }\) (Russian). Dokl. Akad. Nauk. Armyan SSR 83(1), 7–10 (1986)Google Scholar
  30. 30.
    V.K. Tuan, Laplace transform of functions with bounded averages. Int. J. Evol. Equ. 1(4), 429–433 (2005)MathSciNetzbMATHGoogle Scholar
  31. 31.
    V.K. Tuan, A. Boumenir, Sampling in Paley-Wiener and Hardy spaces, §9 in Harmonic, Wavelet and p-adic Analysis, ed. by N.M. Chuong, Y.V. Egorov, M.Y. Khrennikov, D. Mumford (World Scientific Publishing, Hackensack, 2007), pp. 175–209Google Scholar
  32. 32.
    V.K. Tuan, D.T. Duc, Automatic evaluation of abscissa of convergence for inverse Laplace transform. Frac. Calc. Appl. Anal. 3(4), 353–358 (2000)MathSciNetzbMATHGoogle Scholar
  33. 33.
    V.K. Tuan, D.T. Duc, Convergence rate of Post-Widder approximate inversion of the Laplace transform. Vietnam J. Math. 28(1), 93–96 (2000)MathSciNetzbMATHGoogle Scholar
  34. 34.
    V.K. Tuan, D.T. Duc, A new real inversion formula for the Laplace transform and its convergence rate. Frac. Cal. Appl. Anal. 5(4), 387–394 (2002)MathSciNetzbMATHGoogle Scholar
  35. 35.
    V.K. Tuan, N.T. Hong, Interpolation in the Hardy space. Integr. Transf. Spec. Funct. 24(8), 664–671 (2013)MathSciNetCrossRefGoogle Scholar
  36. 36.
    V.K. Tuan, T. Tuan, A real-variable inverse formula for the Laplace transform. Integr. Transf. Spec. Funct. 23(8), 551–555 (2012)MathSciNetCrossRefGoogle Scholar
  37. 37.
    V.K. Tuan, O.I. Marichev, S.B. Yakubovich, Composition structure of integral transformations. Soviet Math. Dokl. 33(1), 166–170 (1986)zbMATHGoogle Scholar
  38. 38.
    F. Veillon, Algorithm 486: numerical inversion of Laplace transform [D5]. Commun. ACM 17(10), 587–589 (1974)CrossRefGoogle Scholar
  39. 39.
    D.V. Widder, The inversion of the Laplace integral and the related moment problem. Trans. Am. Math. Soc. 36, 107–200 (1934)MathSciNetCrossRefGoogle Scholar
  40. 40.
    D.V. Widder, The Laplace Transform (Princeton University Press, Princeton, 1946)Google Scholar
  41. 41.
    S.B. Yakubovich, A real inversion formula for the bilateral Laplace transform (Russian). Izv. Nats. Akad. Nauk Armenii Mat. 40(3), 67–79 (2005)MathSciNetGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of West GeorgiaCarrolltonUSA
  2. 2.Department of MathematicsQuy Nhon UniversityQuy Nhon, Binh DinhViet Nam

Personalised recommendations