Evaluation of classes of definite integrals involving elementary functions via differentiation of special functions

  • K. O. Geddes
  • M. L. Glasser
  • R. A. Moore
  • T. C. Scott
Article

Abstract

Herein, it is shown that by exploiting integral definitions of well known special functions, through generalizations and differentiations, broad classes of definite integrals can be solved in closed form or in terms of special functions. This is especially useful when there is no closed form solution to the indefinite form of the integral. In this paper, three such classes of definite integrals are presented. Two of these classes incorporate and supercede all of Kölbig's integration formulae [11], including his formulation for the computation of Cauchy principal values. Also presented are the mathematical derivations that support the implementation of a third class which exploits the incomplete Gamma function. The resulting programs, based on pattern matching, differentiation, and occasionally limits, are very efficient.

Keywords

Integration Symbolic Computation MAPLE Special Functions 

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References

  1. 1.
    Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions. New York; Dover 1970 (9th printing)Google Scholar
  2. 2.
    Bronstein, Manuel, Integration of Elementary Functions. Berkeley, California: University of California 1987 (Ph.D. thesis)Google Scholar
  3. 3.
    Char, B. W., Geddes, K. O., Gonnet, G. H., Monagan, M. B., Watt, S. M.: Maple Reference Manual, 5th ed. Ontario, Canada: Watcom Waterloo 1988Google Scholar
  4. 4.
    Cherry, G.: Integration in finite terms with special functions: the error function. J. Symb. Comput.1, 283–302 (1985)Google Scholar
  5. 5.
    Cherry, G.: Integration in finite terms with special functions: the logarithmic integral. SIAM J. Comput.15, 1–21 (1986)Google Scholar
  6. 6.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G.: Tables of integral transforms. Vol. I, New York: McGraw-Hill 1954Google Scholar
  7. 7.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G.: Tables of Integral Transforms. Vol. II, New York: McGraw-Hill 1954Google Scholar
  8. 8.
    Geddes, K. O., Scott, T. C.: Recipes for classes of definite integrals involving exponentials and logarithms, In: Proceedings of Computers and Mathematics, pp. 192–201; Kaltofen E., Watt S. M. (eds.), (conference held at MIT) Berlin, Heidelberg, New York: Springer 1989Google Scholar
  9. 9.
    Kölbig, K. S.: Closed expressions for\(\int\limits_0^1 {t^{ - 1} \log ^{n - 1} t\log ^p (1 - t)dt} \). Math. Comp.39, 647–654 (1982)Google Scholar
  10. 10.
    Kölbig, K. S.: On the integral\(\int\limits_0^{\pi /2} {\log ^n \cos x\log ^p \sin xdx} \). Math. Comp.40, 565–570 (1983)Google Scholar
  11. 11.
    Kölbig, K. S.: Explicit evaluation of certain definite integrals involving powers of logarithms. J. Symb. Comput.1, 109–114 (1985)Google Scholar
  12. 12.
    Kölbig, K. S.: On the integral\(\int\limits_0^\infty {e^{ - \mu t} t^{v - 1} \log ^m tdt} \). Math. Comp.41, 171–182 (1985)Google Scholar
  13. 13.
    Kölbig, K. S.: On the integral\(\int\limits_0^\infty {x^{v - 1} (1 + \beta x)^{ - \lambda } \ln ^m xdx} \). J. Comp. Appl. Math.14, 319–344 (1986)Google Scholar
  14. 14.
    Luke, Y. L.: The special functions and their approximations. Vol. 1, New York, London: Academic Press 1969Google Scholar
  15. 15.
    Prudnikov, A. P., Brychkov, Yu. A., Marichev, O. I.: Integrals and series, Gordon & Breach Science, 2nd printing, New York (1988). Vol. 1, Translated from the Russian “Integraly i ryady by Nauka” (1981)Google Scholar
  16. 16.
    Risch, R. H.: The problem of integration in finite terms. Trans. Am. Math. Soc.139, 167–189 (1969)Google Scholar
  17. 17.
    Scott, T. C.: A new approach to definite integration for maple 4.3, Maple Newsletter no. 3 (Symbolic Computation Group), Ontario: Waterloo (1988)Google Scholar
  18. 18.
    Scott, T. C., Moore, R. A., Fee, G. J., Monagan, M. B., Labahn, G., Geddes, K. O.: Perturbative Solutions of Quantum Mechanical Problems by Symbolic Computation: A Review. Int. J. Mod. Phys. C1, 53–76 (1990). (Invitation from World Scientific)Google Scholar
  19. 19.
    Trager, Barry M.: Integration of algebraic functions. Cambridge, MA: Massachusetts Institute of Technology 1984. (Ph.D. thesis)Google Scholar
  20. 20.
    Wang, P.S.: Evaluation of definite integrals by symbolic manipulation. Cambridge, MA: Massachusetts Institute of Technology, 1971. (Ph.D. thesis)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • K. O. Geddes
    • 1
  • M. L. Glasser
    • 2
  • R. A. Moore
    • 3
  • T. C. Scott
    • 3
    • 4
  1. 1.Department of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Department of Physics and Department of Mathematics and Computer ScienceClarkson UniversityPotsdamUSA
  3. 3.Guelph-Waterloo Program for Graduate Work in PhysicsWaterloo CampusUSA
  4. 4.the Institute for Theoretical Atomic and Molecular Physics at the Harvard—Smithsonian Center for AstrophysicsCambridgeUSA

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