×
Epstein, H. I.; Caviness, B. F.

A structure theorem for the elementary functions and its application to the identity problem. (English) Zbl 0421.68043

Int. J. Comput. Inform. Sci. 8, 9-37 (1979).
Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page
scan of Zbl 0421.68043

Cited in 1 Document

MSC:

68W30 Symbolic computation and algebraic computation
13N05 Modules of differentials
12H05 Differential algebra
12F20 Transcendental field extensions
47E05 General theory of ordinary differential operators

Keywords:

canonical form; elementary transcendental functions; exponential function; Liouville fields; logarithm function; structure theorem; fields of elementary functions; algorithms for integration

Software:

MACSYMA; ALTRAN; REDUCE
Cite Review PDF
Full Text: DOI

References:

[1] James Ax, ?On Schanuel’s conjectures,?Ann. Math. 93:252?568 (1971). · Zbl 0232.10026 · doi:10.2307/1970774
[2] W. S. Brown, ?Rational exponential expressions and a conjecture concerning ? ande,?Am. Math. Mon. 76:28?34 (January 1969). · Zbl 0175.33301 · doi:10.2307/2316782
[3] B. F. Caviness, ?On canonical forms and simplification,?J. Ass. Comput. Mach. 17:385?396 (April 1970). · Zbl 0193.31302 · doi:10.1145/321574.321591
[4] B. F. Caviness and Michael Rothstein, ?A Liouville theorem on integration in finite terms for line integrals,?Commun. Alg. 3:781?795 (1975). · Zbl 0339.12106 · doi:10.1080/00927877508822073
[5] G. E. Collins, ?The SAC-1 System: An Introduction and Survey,?Proc. Second Symposium on Symbolic and Algebraic Manipulation, S. R. Petrick, ed. (Association for Computing Machinery, New York, 1971), pp. 144?152.
[6] H. I. Epstein, ?Algorithms for Elementary Transcendental Function Arithmetic,? Ph.D. thesis, University of Wisconsin, Madison, Wisconsin (April 1975), 409 pp. (Available from University Microfilms International, Ann Arbor, Michigan).
[7] R. J. Fateman, ?Essays in Algebraic Simplification,? Project MAC TR-95, Massachusetts Institute of Technology, Cambridge, Mass. (April 1972).
[8] J. P. Fitch, ?On algebraic simplification,?Comput. J. 18:23?27 (February 1973). · Zbl 0251.68022 · doi:10.1093/comjnl/16.1.23
[9] J. H. Griesmer and R. D. Jenks, ?SCRATCHPAD/1-An Interactive Facility for Symbolic Mathematics,?Proc. Second Symposium on Symbolic and Algebraic Manipulation, S. R. Petrick, ed. (Association for Computing Machinery, New York, 1971), pp. 42?58.
[10] A. D. Hall Jr., ?The Altran System for Rational Function Manipulation-A Survey,Proc. Second Symposium on Symbolic and Algebraic Manipulation, S. R. Petrick, ed. (Association for Computing Machinery, New York, 1971).
[11] G. H. Hardy,The Integration of Functions of a Single Variable, 2nd Ed. (Cambridge University Press, London, 1928).
[12] A. C. Hearn, ?REDUCE-2: A System and Language for Algebraic Manipulation,?Proc. Second Symposium on Symbolic and Algebraic Manipulation, S. R. Petrick, ed. (Association for Computing Machinery, New York, 1971), pp. 128?133.
[13] S. C. Johnson, ?On the problem of recognizing zero,?J. Assoc. Comput. Mach. 18:559?565 (October 1971). · Zbl 0229.68012 · doi:10.1145/321662.321669
[14] Irving Kaplansky,An Introduction to Differential Algebra (Hermann, Paris, 1957). · Zbl 0083.03301
[15] E. R. Kolchin, ?Algebraic groups and algebraic dependence,?Am. J. Math. 90:1151?1164 (1968). · Zbl 0169.36701 · doi:10.2307/2373294
[16] E. R. Kolchin,Differential Algebra and Algebraic Groups (Academic Press, New York, 1973). · Zbl 0264.12102
[17] Joseph Liouville, ?Sur la détermination des integrales dont la valeur est algébrique,? J. ÉcolePolytech. 14 (Sect. 23):124?193 (1833).
[18] Joseph Liouville, ?Mémoire sur les transcendantes élliptiques de première et de seconde espèce, considérées comme fonctions de leur amplitude,?J. École Polytech. 14 (Sect. 24):57?83 (1833).
[19] Joseph Liouville, ?Mémoire sur l’intégration d’une classe de fonctions transcendantes,?J. Reine Angew. Math. 13:93?118 (1835). · ERAM 013.0476cj · doi:10.1515/crll.1835.13.93
[20] Joseph Liouville, ?Mémoire sur la classification des transcendantes et sur l’impossibilité d’exprimer les racines de certain équations en fonction finie explicite des coefficients,?J. Math. Pures Appl. 2:56?104 (1837);3:523?546 (1838).
[21] Joseph Liouville, Mémoire sur l’intégration d’une classe d’équations différentielles du second ordre en quantités finies explicites,?J. Math. Pures Appl. 4:423?456 (1839).
[22] Joseph Liouville, ?Mémoire sur les transcendantes élliptiques de première et de seconde espèce, considérées comme fonction de leur module,?J. Math. Pures Appl. 5:34?36, 441?464 (1840).
[23] Joseph Liouville, ?Remarques nouvelles sur l’équation de Riccati,?J. Math. Pures Appl. 6:1?13 (1841).
[24] W. A. Martin and R. J. Fateman, ?The MACSYMA System,?Proc. Second Symposium on Symbolic and Algebraic Manipulation, S. R. Petrick, ed. (Association for Computing Machinery, New York, 1971), pp. 59?75.
[25] Joel Moses, ?Algebraic simplification: a guide for the perplexed,?Commun. ACM 14:527?537 (August 1971). · Zbl 0222.68017 · doi:10.1145/362637.362648
[26] Joel Moses, ?Toward a general theory of special functions,?Commun. ACM 15:550?554 (July 1972). · Zbl 0245.65007 · doi:10.1145/361454.361466
[27] Alexandre Ostrowski, ?Sur les relations algébriques entre les intégrales indéfines,?Acta Math. 78:315?318 (1946). · Zbl 0063.06062 · doi:10.1007/BF02421605
[28] Daniel Richardson. ?Some unsolvable problems involving elementary functions of a real variable,?J. Symb. Logic 33:514?520 (1968). · Zbl 0175.27404 · doi:10.2307/2271358
[29] Daniel Richardson, ?A solution of the identity problem for integral exponential functions,?Z. Math. Logik Grundlagen Math. 15:333?340 (1969). · Zbl 0188.32802 · doi:10.1002/malq.19690152003
[30] Daniel Richardson, ?The simple exponential constant problem,?Z. Math. Logik Grundlagen Math. 17:133?136 (1971). · Zbl 0231.02053 · doi:10.1002/malq.19710170118
[31] R. H. Risch, ?Further Results on Elementary Functions,? IBM Technical Report RC 2402, Yorktown Heights, New York (March 1969).
[32] R. H. Risch, ?The problem of integration in finite terms,?Trans. Am. Math. Soc. 139:167?189 (May 1969). · Zbl 0184.06702 · doi:10.1090/S0002-9947-1969-0237477-8
[33] R. H. Risch, ?The solution of the problem of integration in finite terms,?Bull. Am. Math. Soc. 76:605?608 (May 1970). · Zbl 0196.06801 · doi:10.1090/S0002-9904-1970-12454-5
[34] R. H. Risch, ?Algebraic Properties of the Elementary Functions of Analysis,? IBM Technical Report RC 5565, Yorktown Heights, New York (August 1975).
[35] J. F. Ritt,Integration in Finite Terms: Liouville’s Theory of Elementary Methods (Columbia University Press, New York, 1948). · Zbl 0031.20603
[36] J. F. Ritt,Differential Algebra (American Mathematical Society, Providence, Rhode Island, 1950); republished by Dover Publications, Inc., New York, 1966. · Zbl 0037.18402
[37] Maxwell Rosenlicht, ?Liouville’s theorem on functions with elementary integrals,?Pacific J. Math. 24:153?161 (1968). · Zbl 0155.36702 · doi:10.2140/pjm.1968.24.153
[38] Maxwell Rosenlicht, ?Integration in finite terms,?Am. Math. Mon. 79:963?972 (November 1972). · Zbl 0249.12106 · doi:10.2307/2318066
[39] Maxwell Rosenlicht, ?On theory of elementary functions? (preprint). · Zbl 0318.12107
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.