Numerical Algorithms

, Volume 77, Issue 4, pp 955–982 | Cite as

Detection of the singularities of a complex function by numerical approximations of its Laurent coefficients

  • Mariarosaria Rizzardi
Original Paper


For several applications, it is important to know the location of the singularities of a complex function: just for example, the rightmost singularity of a Laplace Transform is related to the exponential order of its inverse function. We discuss a numerical method to approximate, within an input accuracy tolerance, a finite sequence of Laurent coefficients of a function by means of the Discrete Fourier Transform (DFT) of its samples along an input circle. The circle may also enclose some singularities, since the method works with the Laurent expansion. The DFT is computed by the FFT algorithm so that, from a computational point of view, the efficiency is guaranteed. The function samples may be obtained by solving a numerical problem such as, for example, a differential problem. We derive, as consequences of the method, some new outcomes able to detect those singularities which are close to the circle and to discover if the singularities are all external or internal to the circle so that the Laurent expansion reduces to its regular or singular part, respectively. Other singularities may be located by means of a repeated application of the method, as well as an analytic continuation. Some examples and results, obtained by a first implementation, are reported.


Numerical algorithm Software suite Interactive software Complex singularities Laurent coefficients Analytic continuation 


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  1. 1.
    Bornemann, F.: Accuracy and stability of computing high-order derivatives of analytic functions by cauchy integrals. Found. Comput. Math. 11(1), 1–63 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brent, R.P., Kung, H.T.: Fast algorithms for manipulating formal power series. J. ACM 25(4), 581–595 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Finkston, B.: Cluster data by using the mean shift algorithm.
  4. 4.
    Fukunaga, K., Hostetler, L.D.: The estimation of the gradient of a density function, with applications in pattern recognition. IEEE Trans IT 21(1), 32–40 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Duffy, D.G.: Transform methods for solving partial differential equations. Chapman & Hall/CRC, Florida (2004)CrossRefzbMATHGoogle Scholar
  6. 6.
    Giunta, G., Laccetti, G., Rizzardi, M.: A numerical method for locating the abscissa of convergence of a laplace transform function with no singularity at infinity. Int. J. Comp. Math. 37(1), 89–103 (1990)CrossRefzbMATHGoogle Scholar
  7. 7.
    Gonnet, P., Guettel, S., Trefethen, L.N.: Robust Padé approximation via SVD. SIAM Rev. 55(1), 101–117 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gonnet, P., Pachon, R., Trefethen, L.N.: Robust rational interpolation and least squares. ETNA 38, 146–167 (2011)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Henrici, P.: Applied and computational complex analysis, vol I. Wiley, New York (1988)zbMATHGoogle Scholar
  10. 10.
    Henrici, P.: Applied and computational complex analysis, vol III. Wiley, New York (1993)zbMATHGoogle Scholar
  11. 11.
    Lyness, J.N.: Differentiation formulas for analytic functions. Math. Comp. 22, 352–362 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lyness, J.N., Giunta, G.: A modification of the weeks method for numerical inversion of the laplace transform. Math. Comp. 47, 313–322 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lyness, J.N., Moler, C.B.: Numerical differentiation of analytic functions. SIAM J. Numer. Anal. 4(2), 201–210 (1967)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Lyness, J.N., Sande, G.: Algorithm 413: ENTCAF and ENTCRE. Evaluation of normalized taylor coefficients of an analytic function. Comm. ACM 14, 669–675 (1971)CrossRefGoogle Scholar
  15. 15.
    Needham, T.: Visual complex analysis. Clarendon Press, Oxford (1999)Google Scholar
  16. 16.
    Rizzardi, M.: LaurentCoeffs. (2017)
  17. 17.
    Talbot, A.: The accurate numerical inversion of laplace transforms. J. Inst. Math. Appl. 23, 97–120 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tourigny, Y., Grinfeld, M.: Deciphering singularities by discrete methods. Math. Comp. 62(205), 155–169 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Weeks, T.: Numerical inversion of laplace transform using laguerre functions. J. ACM, 13 (1966)Google Scholar
  20. 20.
    Wegert, E.: Visual complex functions: an introduction with phase portraits. Birkhäuser/Springer (2012)Google Scholar
  21. 21.
    Wegert, E.: Phase plots of complex functions, (upyeard 2014) (2013)
  22. 22.
  23. 23.
    Weideman, J.A.C.: Computing the dynamics of complex singularities of nonlinear PDEs. SIAM J. Appl. Dynam. Systems 2(2), 171–186 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.DiST - Dipartimento di Scienze e TecnologieUniversità degli Studi di Napoli ParthenopeNaplesItaly

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