A conjecture on integration in finite terms with elementary functions and polylogarithms

ABSTRACT

In this abstract, we report on a conjecture that gives the form of an integral if it can be expressed using elementary functions and polylogarithms. The conjecture is proved by the author in the cases of the dilogarithm and the trilogarithm [3] and consists of a generalization of Liouville's theorem on integration in finite terms with elementary functions. Those last structure theorems, for the dilogarithm and the trilogarithm, are the first case of structure theorems where logarithms can appear with non-constant coefficients. In order to prove the conjecture for higher polylogarithms we need to find the functional identities, for the polylogarithms that we are using, that characterize all the possible algebraic relations among the considered polylogarithms of functions that are built up from the rational functions by taking the considered polylogarithms, exponentials, logarithms and algebraics. The task of finding those functional identities seems to be a difficult one and is an unsolved problem for the most part to this date.

REFERENCES

• 1: J. Baddoura, A Dilogarithmic Extension of Liouville's Theorem on Integration in Finite Terms. Technical report submitted to the Air Force Office of Scientific Research, AFOSR-85-0264, ADA 1910149, TR-87-2039, 1987.
• 2: J. Baddoura, Integration in finite terms and simplification with dilogarithms: a progress report, Proceedings of the third conference on Computers and mathematics, p.166-171, May 1989, Boston, Massachusetts, United States
• 3: Mohamed Jamil Baddoura, Integration in Finite Terms with Elementary Functions and Dilogarithms, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 1993.
• 4: R. Coleman, Dilogarithms, Regulators and p-adic L- functions, invent iones Mathemat icao (1982), 171-208.
• 5: L. Lewin, Polylogarithms and Associated Functions, North Holland, New York, NY, 1981.
• 6: Joel Moses, Towards a general theory of special functions, Communications of the ACM, v.15 n.7, p.550-554, July 1972 [doi>10.1145/361454.361466]
• 7: M. Rosenlicht, Liouville's Theorem on Functions with Elementary IntegrMs, Pacific J. Math. vol. 24, No. 1 (1968), 153-161.
• 8: H.F. Sandhaxn, A Logarithmic Transcendent, J. London Math. Soc. vol. 24, 83--91, 1949.
• 9: M. Singer, B. Saunders and B.F. Caviness, An Extension of Liouville's Theorem on Integration in Finite Terms, SIAM J. on Computing (1985), 966-990.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Nonalgebraic algorithms

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS

General Terms:
Algorithms, Theory, Verification