Advertisement

Generalization of Risch’s Algorithm to Special Functions

Chapter
Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)

Abstract

Symbolic integration deals with the evaluation of integrals in closed form. We present an overview of Risch’s algorithm including recent developments. The algorithms discussed are suited for both indefinite and definite integration. They can also be used to compute linear relations among integrals and to find identities for special functions given by parameter integrals. The aim of this presentation is twofold: to introduce the reader to some basic ideas of differential algebra in the context of integration and to raise awareness in the physics community of computer algebra algorithms for indefinite and definite integration.

Notes

Acknowledgements

The author was supported by the Austrian Science Fund (FWF), grant no. W1214-N15 project DK6, by the strategic program “Innovatives OÖ 2010 plus” of the Upper Austrian Government, by DFG Sonderforschungsbereich Transregio 9 “Computergestützte Theoretische Teilchenphysik”, and by the Research Executive Agency (REA) of the European Union under the Grant Agreement number PITN-GA-2010-264564 (LHCPhenoNet).

References

  1. 1.
    Ablinger, J., Blümlein, J., Raab, C.G., Schneider, C., Wißbrock, F.: DESY 13-063 (preprint)Google Scholar
  2. 2.
    Ablinger, J., Blümlein, J., Raab, C.G., Schneider, C.: Iterated Binomial Sums (in preparation)Google Scholar
  3. 3.
    Abramov, S.A.: Rational solutions of linear differential and difference equations with polynomial coefficients. USSR Comput. Math. Math. Phys. 29(6), 7–12 (1989). (English translation of Zh. vychisl. Mat. mat. Fiz. 29, pp. 1611–1620, 1989)Google Scholar
  4. 4.
    Abramov, S.A.: On d’Alembert substitution. In: Proceedings of ISSAC’93, Kiev, pp. 20–26 (1993)Google Scholar
  5. 5.
    Abramov, S.A., Petkovšek, M.: D’Alembertian solutions of linear differential and difference equations. In: Proceedings of ISSAC’94, Oxford, pp. 169–174 (1994)Google Scholar
  6. 6.
    Almkvist, G., Zeilberger, D.: The method of differentiating under the integral sign. J. Symb. Comput. 10, 571–591 (1990)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Baddoura, J.: Integration in finite terms with elementary functions and dilogarithms. J. Symb. Comput. 41, 909–942 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Boettner, S.T.: Mixed transcendental and algebraic extensions for the Risch-Norman algorithm. PhD thesis, Tulane University, New Orleans (2010)Google Scholar
  9. 9.
    Bronstein, M.: A unification of Liouvillian extensions. Appl. Algebra Eng. Commun. Comput. 1, 5–24 (1990)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bronstein, M.: Integration of elementary functions. J. Symb. Comput. 9, 117–173 (1990)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bronstein, M.: On solutions of linear ordinary differential equations in their coefficient field. J. Symb. Comput. 13, 413–439 (1992)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Bronstein, M.: Symbolic Integration Tutorial. Course Notes of an ISSAC’98 Tutorial, Rostock. Available at http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac98.pdf (1998)
  13. 13.
    Bronstein, M.: Symbolic Integration I – Transcendental Functions, 2nd edn. Springer, Berlin/New York (2005)MATHGoogle Scholar
  14. 14.
    Bronstein, M.: Structure theorems for parallel integration. J. Symb. Comput. 42, 757–769 (2007)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Brown, F.C.S.: The massless higher-loop two-point function. Commun. Math. Phys. 287, 925–958 (2009). arXiv:0804.1660Google Scholar
  16. 16.
    Chyzak, F.: An extension of Zeilberger’s fast algorithm to general holonomic functions. Discret. Math. 217, 115–134 (2000)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Chyzak, F., Kauers, M., Salvy, B.: A non-holonomic systems approach to special function identities. In: Proceedings of ISSAC’09, Seoul, pp. 111–118 (2009)Google Scholar
  18. 18.
    Czichowski, G.: A note on Gröbner bases and integration of rational functions. J. Symb. Comput. 20, 163–167 (1995)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Gradshteyn, I.S., Ryzhik, I.M.: In: Jeffrey, A., Zwillinger, D. (eds.) Table of Integrals, Series, and Products, 7th edn. Academic, New York (2007)Google Scholar
  20. 20.
    Gröbner, W., Hofreiter, N.: Integraltafel – Zweiter Teil: Bestimmte Integrale, 4th edn. Springer, Wien (1966)Google Scholar
  21. 21.
    Hermite, C.: Sur l’intégration des fractions rationelles. Nouvelles annales de mathématiques (2e série) 11, 145–148 (1872)Google Scholar
  22. 22.
    Kaplansky, I.: An Introduction to Differential Algebra. Hermann, Paris (1957)MATHGoogle Scholar
  23. 23.
    Kauers, M.: Integration of algebraic functions: a simple heuristic for finding the logarithmic part. In: Proceedings of ISSAC’08, Hagenberg, pp. 133–140 (2008)Google Scholar
  24. 24.
    Koutschan, C.: Advanced applications of the holonomic systems approach. PhD thesis, Johannes Kepler Universität Linz (2009)Google Scholar
  25. 25.
    Lazard, D., Rioboo, R.: Integration of rational functions: rational computation of the logarithmic part. J. Symb. Comput. 9, 113–115 (1990)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Liouville, J.: Premier mémoire sur la detérmination des intégrales dont la valeur est algébrique. J. l’Ecole Polytechnique 14, 124–148 (1833)Google Scholar
  27. 27.
    Liouville, J.: Second mémoire sur la detérmination des intégrales dont la valeur est algébrique. J. l’Ecole Polytechnique 14, 149–193 (1833)Google Scholar
  28. 28.
    Liouville, J.: Mémoire sur l’intégration d’une classe de fonctions transcendantes. J. reine und angewandte Mathematik 13, 93–118 (1835)CrossRefMATHGoogle Scholar
  29. 29.
    Mack, C.: Integration of affine forms over elementary functions. Computational Physics Group Report UCP-39, University of Utah (1976)Google Scholar
  30. 30.
    Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics. Springer, Berlin (1966)MATHGoogle Scholar
  31. 31.
    Mulders, T.: A note on subresultants and the Lazard/Rioboo/Trager formula in rational function integration. J. Symb. Comput. 24, 45–50 (1997)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Norman, A.C.: Integration in Finite Terms. In: Buchberger, B., Collins, G.E., Loos, R., Albrecht R. (eds.) Computer Algebra: Symbolic and Algebraic Computation, pp. 57–69. Springer, Wien (1983)CrossRefGoogle Scholar
  33. 33.
    Norman, A.C., Moore, P.M.A.: Implementing the new Risch Integration algorithm. In: Proceedings of the 4th International Colloquium on Advanced Computing Methods in Theoretical Physics, Saint-Maximin, pp. 99–110 (1977)Google Scholar
  34. 34.
    Piquette, J.C.: A method for symbolic evaluation of indefinite integrals containing special functions or their products. J. Symb. Comput. 11, 231–249 (1991)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Piquette, J.C., Van Buren, A.L.: Technique for evaluating indefinite integrals involving products of certain special functions. SIAM J. Math. Anal. 15, 845–855 (1984)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, vols. 1–3. Gordon & Breach, New York (1986–1990)Google Scholar
  37. 37.
    Raab, C.G.: Definite integration in differential fields. PhD thesis, JKU Linz (2012)Google Scholar
  38. 38.
    Raab, C.G.: Using Gröbner bases for finding the logarithmic part of the integral of transcendental functions. J. Symb. Comput. 47, 1290–1296 (2012)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Remiddi, E., Vermaseren, J.A.M.: Harmonic polylogarithms. Int. J. Mod. Phys. A15, 725–754 (2000). arXiv:hep-ph/9905237Google Scholar
  40. 40.
    Rich, A.D., Jeffrey, D.J.: A knowledge repository for indefinite integration based on transformation rules. In: Proceedings of Calculemus/MKM 2009, Grand Bend, pp. 480–485 (2009)Google Scholar
  41. 41.
    Risch, R.H.: The problem of integration in finite terms. Trans. Am. Math. Soc. 139, 167–189 (1969)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Ritt, J.F.: Integration in Finite Terms – Liouville’s Theory of Elementary Methods. Columbia University Press, New York (1948)MATHGoogle Scholar
  43. 43.
    Rothstein, M.: Aspects of symbolic integration and simplification of exponential and primitive functions. PhD thesis, Univ. of Wisconsin-Madison (1976)Google Scholar
  44. 44.
    Singer, M.F.: Liouvillian solutions of linear differential equations with Liouvillian coefficients. J. Symb. Comput. 11, 251–273 (1991)CrossRefMATHGoogle Scholar
  45. 45.
    Singer, M.F., Saunders, B.D., Caviness, B.F.: An extension of Liouville’s theorem on integration in finite terms. SIAM J. Comput. 14, 966–990 (1985)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Trager, B.M.: Algebraic factoring and rational function integration. In: Proceedings of SYMSAC’76, Yorktown Heights, pp. 219–226 (1976)Google Scholar
  47. 47.
    Zeilberger, D.: A holonomic systems approach to special functions identities. J. Comp. Appl. Math. 32, 321–368 (1990)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Zeilberger, D.: A fast algorithm for proving terminating hypergeometric identities. Discret. Math. 80, 207–211 (1990)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Deutsches Elektronen-Synchrotron, DESYZeuthenGermany

Personalised recommendations