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Integration — What do we want from the theory?

  • J. H. Davenport
Algorithms 1 — Miscellaneous
Part of the Lecture Notes in Computer Science book series (LNCS, volume 162)

Abstract

The theory of integration has moved a long way in the last fourteen years, though not far enough to satisfy the demands placed on it by its customers. This paper outlines what problems have yet to be solved, and tries to explain why they are not trivial.

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Bibliography

  1. Davenport, 1981.
    Davenport, J.H., On the Integration of Algebraic Functions. Springer Lecture Notes in Computer Science 102, Springer-Verlag, Berlin-Heidelberg-New York, 1981. Zbl. 471.14009.Google Scholar
  2. Davenport, 1982.
    Davenport, J.H., On the Parallel Risch Algorithm (I). Proc. EUROCAM 82 (Springer Lecture Notes in Computer Science, Springer-Verlag, Berlin-Heidelberg-New York, 144, 1982) pp. 144–157.Google Scholar
  3. Davenport, 1983.
    Davenport, J.H., Les Equations Différentielles Ordinaires de Risch sur une Courbe Algébrique. To appear (and I.M.A.G. Research Report, April 1983).Google Scholar
  4. Hardy, 1916.
    Hardy,G.H., The Integration of Functions of a Single Variable (2nd. ed.). Cambridge Tract 2, C.U.P., 1916.Google Scholar
  5. Hermite. 1972
    Hermite, E., Sur l'intégration des fractions rationnelles. Nouvelles Annales de Mathématiques, 2 Sér., 11(1872) pp. 145–148. Annales Scientifiques de l'Ecole Normale Supérieure, 2 Sér., 1(1872) pp. 215–218.Google Scholar
  6. Loos, 1982.
    Loos, R., Computing in Algebraic Extensions. Computing Supplementum 4 (ed. B. Buchberger et al.), Springer-Verlag, Wien-New York, 1982, pp. 173–187.Google Scholar
  7. Moses & Zippel, 1979.
    Moses, J. & Zippel, R.E., An Extension of Liouville's Theorem. Proc. EUROSAM 79 (Springer Lecture Notes in Computer Science 72, Springer-Verlag, Berlin-Heidelberg-New York, 1979) pp. 426–430.Google Scholar
  8. Norman, 1982.
    Norman, A.C., Integration in Finite Terms. Computing Supplementum 4 (ed. B. Buchberger et al.), Springer-Verlag, Wien-New York, 1982, pp. 57–69.Google Scholar
  9. Risch, 1969.
    Risch, R.H., The Problem of Integration in Finite Terms. Trans. A.M.S. 139(1969) pp. 167–189. MR 38(1969) #5759. Zbl. 184,67.Google Scholar
  10. Risch, 1970.
    Risch, R.H., The Solution of the Problem of Integration in Finite Terms. Bull. A.M.S. 76(1970) pp. 605–608. MR 42(1971) #4530.Google Scholar
  11. Rothstein, 1976.
    Rothstein, M., Aspects of Symbolic Integration and Simplification of Exponential and Primitive Functions. Ph.D. Thesis, University of Wisconsin, Madison, 1976 (Xerox University Microfilms 77-8809).Google Scholar
  12. Rothstein, 1977.
    Rothstein, M., A New Algorithm for the Integration of Exponential and Logarithmic Functions. Proc. 1977 MACSYMA Users' Conference (NASA Publication CP-2012, National Technical Information Service, Springfield, Virginia, 1977) pp.263–274.Google Scholar
  13. Rothstein & Caviness, 1979.
    Rothstein, M. & Caviness, B.F., A Structure Theorem for Exponential and Primitive Functions. SIAM J. Comp 8(1979) pp. 357–367. MR 80m:12028.Google Scholar
  14. Singer et al., 1981
    Singer, M.F., Saunders, B.D. & Caviness, B.F., An Extension of Liouville's Theorem on Integration in Finite Terms. Proc. SYMSAC 81 (ACM, New York, 1981), pp. 23–24. Zbl. 482.12008.Google Scholar
  15. Trager, 1976.
    Trager, B.M., Algebraic Factoring and Rational Function Integration. Proc. SYMSAC 76 (ed. R.D. Jenks), ACM. New York, 1976, pp. 219–226.Google Scholar
  16. Trager, 1979.
    Trager, B.M., Integration of Simple Radical Extensions. Proc. EUROSAM 79 (Springer Lecture Notes in Computer Science 72, Springer-Verlag, Berlin-Heidelberg-New York, 1979) pp. 408–414. MR 82j:12032.Google Scholar
  17. Winograd, 1979.
    Winograd, S., On Multiplication in Algebraic Extension Fields. Theor. Comp. Sci. 8(1979) pp. 359–377.Google Scholar
  18. Winograd, 1980.
    Winograd, S., On the Multiplication of Polynomials Modulo a Polynomial. SIAM J. Comp. 9(1980) pp. 225–229. Zbl. 446.68032.Google Scholar
  19. Yun, 1977.
    Yun, D.Y.Y., Fast Algorithm for Rational Function Integration. Proc. IFIP 77 (ed. B. Gilchrist), North-Holland, Amsterdam-New York-Oxford. 1977. pp. 493–498 [IBM Research Report RC 6563 6th. January 1977].Google Scholar
  20. Zejli, 1983.
    Zejli,H., Private Communication.Google Scholar
  21. Zolotarev. 1874
    Zolotarev,E., Sur la méthode d'intégration de M. Tchebycheff. Math. Annalen Band V Sér 560. Journal de Mathématiques 2 Sér. 19(1874).Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • J. H. Davenport
    • 1
  1. 1.Equipe d'Analyse NumériqueLaboratoire I.M.A.G.Saint Martin d'HèresFrance

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