Univariate power series expansions in algebraic manipulation

ABSTRACT

In this paper we present a complete algorithm for the determination of univariate power series expansions of meromorphic functions on a Riemann surface. The difficulties involved when expanding at singularities of various forms are discussed. We demonstrate how to use these techniques to calculate limits and as an aid in solving polynomial equations. Finally we discuss several of the implementations of power series manipulation systems with special emphasis on the implementation in MACSYMA.

REFERENCES

• 1: ALTRAN Users Manual, Bell Telephone Labs, Murray Hill, N.J., 1973
• 2: Chang, Y.F., "Automatic Solution of Differential Equations," in Lecture Notes in Math. no 430, Springer Verlag, New York, 1975.
• 3: Euler, B. Introductio in Analysin Infinitorium, Lausanne, 1748.
• 4: Euler, B. Einleitung in die Analysis des Unendlichen, aus dem Lateinischen ubersetzt von J.A.C.Michelsen, Berlin, 2 Vols, 1788.
• 5: Forsyth, A.R., Theory of Functions of a Complex Variable, Cambridge Univ. Press, 1918.
• 6: Forsyth, A.R., Theory of functions of a Complex Variable, reprint of {5}, Dover, New York, 1965.
• 7: W. Morven Gentleman, Experience with truncated power series, ACM SIGSAM Bulletin, v.8 n.3, p.61-62, August 1974 [doi>10.1145/1086837.1086846]
• 8: Gould, "Coefficient Identities for power of Taylor and Dirichlet Series," American Mathematical Monthy, vol 81, pp 3-14, 1974.
• 9: von Holdt, "Rational powers of power series," American Mathematical Monthy, vol 72, pp 740-743, 1965.
• 10: Donald E. Knuth, The art of computer programming, volume 1 (3rd ed.): fundamental algorithms, Addison Wesley Longman Publishing Co., Inc., Redwood City, CA, 1997
• 11: Donald E. Knuth, The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms, Addison-Wesley Longman Publishing Co., Inc., Boston, MA, 1997
• 12: Laurent, J-P., "A Heuristic Program to Compute Limits," Artificial Intelligence, vol 4, 69-94, 1973.
• 13: MACSYMA Reference Manual, Mathlab Group, Project MAC, M.I.T., Cambridge, Mass., November 1975.
• 14: John McCarthy, LISP 1.5 Programmer's Manual, The MIT Press, 1962
• 15: Arthur C. Norman, The SCRATCHPAD power series package, ACM SIGSAM Bulletin, v.9 n.1, p.13-20, February 1975 [doi>10.1145/1088295.1088298]
• 16: Olson, A.M and Ball, W.E., An Investigation of Some Extensions of a Method for Generating Power Series, Computer Systems Lab., Washington University Tech Report No. 12, St. Louis, Mo., April 1969.
• 17: Phragmin, Acta Mathematica, Tome viii, pp 33-42, 1885.
• 18: Picard, "Memoire sur les fonctions entieres," Annales de l'Ecole Normate Superieur, 2me Ser., tome ix, 1880, pp 145-166.
• 19: Weierstrass, Mathematische Werke, Mayer and Muller, Berlin, 1895.
• 20: Richard Zippel, A MACSYMA solution to problem 8, ACM SIGSAM Bulletin, v.9 n.1, p.10-12, February 1975 [doi>10.1145/1088295.1088297]
• 21: Zippel, R.E., "Multivariate Power Series Expansions," in preparation.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.0 General

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.5 Roots of Nonlinear Equations
          Subjects: Systems of equations

General Terms:
Algorithms, Theory