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A new algorithm for computing symbolic limits using hierarchical series

  • Keith O. Geddes
  • Gaston H. Gonnet
Arithmetic
Part of the Lecture Notes in Computer Science book series (LNCS, volume 358)

Abstract

We describe an algorithm for computing symbolic limits, i.e. limits of expressions in symbolic form, using hierarchical series. A hierarchical series consists of two levels: an inner level which uses a simple generalization of Laurent series with finite principal part and which captures the behaviour of subexpressions without essential singularities, and an outer level which captures the essential singularities. Once such a series has been computed for an expression at a given point, the limit of the expression at the point is determined by looking at the most significant term of the series. By this method one can compute, for example, the limit of:
$$\mathop {\lim }\limits_{n \to \infty } \frac{{e^{e^{\sqrt {\ln \ln n} } } }}{{e^{\sqrt {\ln n} } }} - n^{\sqrt {\ln n} } .$$
This algorithm solves the limit problem for a large class of expressions.

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References

  1. 1.
    Paul S. Wang, Automatic Computation of Limits, pp. 458–464 in Proceedings of the Second Symposium on Symbolic and Algebraic Manipulation, ed. S.R. Petrick, Special Interest Group on Symbolic and Algebraic Manipulation, Association for Computing Machinery (1971).Google Scholar
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    W.A. Martin and R.J. Fateman, The MACSYMA System, pp. 59–75 in Proceedings of the Second Symposium on Symbolic and Algebraic Manipulation, ed. S.R. Petrick, Special Interest Group on Symbolic and Algebraic Manipulation, Association for Computing Machinery (1971).Google Scholar
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    Bruce W. Char, Gregory J. Fee, Keith O. Geddes, Gaston H. Gonnet, and Michael B. Monagan, A tutorial introduction to Maple, Journal of Symbolic Computation 2(2) pp. 179–200 (1986).Google Scholar
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    R.H. Risch, The problem of integration in finite terms, Trans. Am. Math. Soc. 139 pp. 167–189 (1969).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Keith O. Geddes
    • 1
  • Gaston H. Gonnet
    • 1
  1. 1.Symbolic Computation Group Department of Computer ScienceUniversity of WaterlooWaterlooCanada

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