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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ablinger, J., Blümlein, J., Raab, C.G., Schneider, C., Wißbrock, F.: DESY 13-063 (preprint)
Ablinger, J., Blümlein, J., Raab, C.G., Schneider, C.: Iterated Binomial Sums (in preparation)
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)
Abramov, S.A.: On d’Alembert substitution. In: Proceedings of ISSAC’93, Kiev, pp. 20–26 (1993)
Abramov, S.A., Petkovšek, M.: D’Alembertian solutions of linear differential and difference equations. In: Proceedings of ISSAC’94, Oxford, pp. 169–174 (1994)
Almkvist, G., Zeilberger, D.: The method of differentiating under the integral sign. J. Symb. Comput. 10, 571–591 (1990)
Baddoura, J.: Integration in finite terms with elementary functions and dilogarithms. J. Symb. Comput. 41, 909–942 (2006)
Boettner, S.T.: Mixed transcendental and algebraic extensions for the Risch-Norman algorithm. PhD thesis, Tulane University, New Orleans (2010)
Bronstein, M.: A unification of Liouvillian extensions. Appl. Algebra Eng. Commun. Comput. 1, 5–24 (1990)
Bronstein, M.: Integration of elementary functions. J. Symb. Comput. 9, 117–173 (1990)
Bronstein, M.: On solutions of linear ordinary differential equations in their coefficient field. J. Symb. Comput. 13, 413–439 (1992)
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)
Bronstein, M.: Symbolic Integration I – Transcendental Functions, 2nd edn. Springer, Berlin/New York (2005)
Bronstein, M.: Structure theorems for parallel integration. J. Symb. Comput. 42, 757–769 (2007)
Brown, F.C.S.: The massless higher-loop two-point function. Commun. Math. Phys. 287, 925–958 (2009). arXiv:0804.1660
Chyzak, F.: An extension of Zeilberger’s fast algorithm to general holonomic functions. Discret. Math. 217, 115–134 (2000)
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)
Czichowski, G.: A note on Gröbner bases and integration of rational functions. J. Symb. Comput. 20, 163–167 (1995)
Gradshteyn, I.S., Ryzhik, I.M.: In: Jeffrey, A., Zwillinger, D. (eds.) Table of Integrals, Series, and Products, 7th edn. Academic, New York (2007)
Gröbner, W., Hofreiter, N.: Integraltafel – Zweiter Teil: Bestimmte Integrale, 4th edn. Springer, Wien (1966)
Hermite, C.: Sur l’intégration des fractions rationelles. Nouvelles annales de mathématiques (2e série) 11, 145–148 (1872)
Kaplansky, I.: An Introduction to Differential Algebra. Hermann, Paris (1957)
Kauers, M.: Integration of algebraic functions: a simple heuristic for finding the logarithmic part. In: Proceedings of ISSAC’08, Hagenberg, pp. 133–140 (2008)
Koutschan, C.: Advanced applications of the holonomic systems approach. PhD thesis, Johannes Kepler Universität Linz (2009)
Lazard, D., Rioboo, R.: Integration of rational functions: rational computation of the logarithmic part. J. Symb. Comput. 9, 113–115 (1990)
Liouville, J.: Premier mémoire sur la detérmination des intégrales dont la valeur est algébrique. J. l’Ecole Polytechnique 14, 124–148 (1833)
Liouville, J.: Second mémoire sur la detérmination des intégrales dont la valeur est algébrique. J. l’Ecole Polytechnique 14, 149–193 (1833)
Liouville, J.: Mémoire sur l’intégration d’une classe de fonctions transcendantes. J. reine und angewandte Mathematik 13, 93–118 (1835)
Mack, C.: Integration of affine forms over elementary functions. Computational Physics Group Report UCP-39, University of Utah (1976)
Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics. Springer, Berlin (1966)
Mulders, T.: A note on subresultants and the Lazard/Rioboo/Trager formula in rational function integration. J. Symb. Comput. 24, 45–50 (1997)
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)
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)
Piquette, J.C.: A method for symbolic evaluation of indefinite integrals containing special functions or their products. J. Symb. Comput. 11, 231–249 (1991)
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)
Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, vols. 1–3. Gordon & Breach, New York (1986–1990)
Raab, C.G.: Definite integration in differential fields. PhD thesis, JKU Linz (2012)
Raab, C.G.: Using Gröbner bases for finding the logarithmic part of the integral of transcendental functions. J. Symb. Comput. 47, 1290–1296 (2012)
Remiddi, E., Vermaseren, J.A.M.: Harmonic polylogarithms. Int. J. Mod. Phys. A15, 725–754 (2000). arXiv:hep-ph/9905237
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)
Risch, R.H.: The problem of integration in finite terms. Trans. Am. Math. Soc. 139, 167–189 (1969)
Ritt, J.F.: Integration in Finite Terms – Liouville’s Theory of Elementary Methods. Columbia University Press, New York (1948)
Rothstein, M.: Aspects of symbolic integration and simplification of exponential and primitive functions. PhD thesis, Univ. of Wisconsin-Madison (1976)
Singer, M.F.: Liouvillian solutions of linear differential equations with Liouvillian coefficients. J. Symb. Comput. 11, 251–273 (1991)
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)
Trager, B.M.: Algebraic factoring and rational function integration. In: Proceedings of SYMSAC’76, Yorktown Heights, pp. 219–226 (1976)
Zeilberger, D.: A holonomic systems approach to special functions identities. J. Comp. Appl. Math. 32, 321–368 (1990)
Zeilberger, D.: A fast algorithm for proving terminating hypergeometric identities. Discret. Math. 80, 207–211 (1990)
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).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Wien
About this chapter
Cite this chapter
Raab, C.G. (2013). Generalization of Risch’s Algorithm to Special Functions. In: Schneider, C., Blümlein, J. (eds) Computer Algebra in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1616-6_12
Download citation
DOI: https://doi.org/10.1007/978-3-7091-1616-6_12
Published:
Publisher Name: Springer, Vienna
Print ISBN: 978-3-7091-1615-9
Online ISBN: 978-3-7091-1616-6
eBook Packages: Computer ScienceComputer Science (R0)