Fast Computation of Some Special Integrals of Mathematical Physics

  • Ekatherina A. Karatsuba


The application of the FEE method to the fast calculation of the values of some special integrals of mathematical physics, such as the probability integral, the Fresnel integrals, integral sine, cosine etc. is considered. The computational complexity is near to optimal.


Steklov Institute Fast Computation Scientific Computing Riemann Zeta Function Interval Method 
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  1. [1]
    E. Baca, The complexity of number-theoretic constants. Info. Proc. Letters 62 pp. 145–152 (1997).Google Scholar
  2. [2]
    Benderskij Yu. V., Fast Computations. Dokl. Akad. Nauk SSSR v 223 5 pp. 1041–1043 (1975).Google Scholar
  3. [3]
    B. C. Berndt Ramanujan’s Notebook, Part I. Springer-Verlag (1985).Google Scholar
  4. [4]
    J. M. Borwein and P. B. Borwein, Pi and the AGM. Wiley, New York(1987).Google Scholar
  5. [5]
    D. M. Bradley, A Class of Series Acceleration Formulae for Catalan’s Constant. The Ramanujan Journal 3 pp. 159–173 (1999).zbMATHCrossRefGoogle Scholar
  6. [6]
    R. P. Brent, Fast Multiple-Precision Evaluation of Elementary Functions. ACM v. 23 2 pp. 242–251 (1976).MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    S. A. Coox, On the minimum computation time of functions. Thesis, Harvard University (1966).Google Scholar
  8. [8]
    Karatsuba A. AND Ofman Yu. , Multiplication of Multiplace Numbers on Automata. Dokl. Akad. Nauk SSSR v. 145 2 pp. 293–294(1962).Google Scholar
  9. [9]
    A. A. Karatsuba, The Complexity of Computations. Proceedings of the Steklov Institute of Mathematics v. 211 pp. 169–183 (1995).Google Scholar
  10. [10]
    E. A. Karatsuba, Fast evaluations of transcendental functions. Probi. Peredachi Inf. 27 (4) pp. 87–110 (1991).MathSciNetGoogle Scholar
  11. [11]
    E. A. Karatsuba, Fast calculation of ((3). Problems of Inform. Transmission 29 (1) pp. 58–62 (1993).MathSciNetzbMATHGoogle Scholar
  12. [12]
    Catherine A. Karatsuba, Fast evaluation of Bessel functions. Integral Transforms and Special Functions 1 (4) pp. 269–276 (1993).MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    E. A. Karatsuba, Fast calculation of the Riemann zeta function (s) for integer values of the argument s. Problems of Information Transmission 31 (4) pp. 353–362 (1995).MathSciNetzbMATHGoogle Scholar
  14. [14]
    E. A. Karatsuba, Fast computation of the values of Hurwitz zeta function and Dirichlet L-series. Problems of Information Transmission 34 (4) pp. 342–353 (1998).MathSciNetzbMATHGoogle Scholar
  15. [15]
    E. A. Karatsuba, Fast evaluation of hypergeometric function by FEE. Proceedings of the thirdCMFT Conference (N. Papamichael, St. Ruscheweyh, E. B. Saff, eds. ) World Scientific pp. 303–314 (1999).Google Scholar
  16. [16]
    E. A. Karatsuba, On the computation of the Euler constant gamma. Numerical Algorithms 24 pp. 83–97 (2000).MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    D. E. Knuth, The art of computer programming. v. 2 Addison-Wesley Publ. Co., Reading (1969).zbMATHGoogle Scholar
  18. [18]
    P. Lindqvist And J. Peetre, On the Remainder in a Series of Mertens. Expo. Math. 15 pp. 467–478 (1997).MathSciNetzbMATHGoogle Scholar
  19. [19]
    N. Nielsen, Theorie des Integrallogarithmus und verwandter Transcendenten. Leipzig, Teubner (1906).Google Scholar
  20. [20]
    Schönhage A. Und Strassen V., Schnelle Multiplikation grosser Zahlen. Computing v. 7 pp. 281–292 (1971).zbMATHCrossRefGoogle Scholar
  21. [21]
    Schonhage A., Grotefeld A. F. W And Vetter E., Fast Algorithms. BIWiss. -Verl. Zürich (1994).Google Scholar
  22. [22]
    C. L. Siegel, Transcendental numbers. Princeton University Press, Princeton (1949).Google Scholar
  23. [23]
    N. M. Temme, Special Functions. Wiley, New York (1996).zbMATHCrossRefGoogle Scholar
  24. [24]
    A. L. Toom, The complexity of a scheme of functional elements realising the multiplication of integers. Dokl. Akad. Nauk SSSR v. 150 3 pp. 496–498 (1963).MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Ekatherina A. Karatsuba
    • 1
  1. 1.Computer Centre of RASMoscowRussia

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