Applications of Some Formulae by Hermite to the Approximation of Exponentials and Logarithms

To C. L. Siegel on his 70th birthday
  • K. Mahler

Abstract

While Liouville gave the first examples of transcendental numbers, the modern theory of proofs of transcendency started with Hermite’s beautiful paper «Sur la fonction exponentielle» (Hermite, 1873). In this paper, for a given system of distinct complex numbers ω 0, ω 1,..., ω m and of positive integers ϱ 0, ϱ 1, ..., ϱ m em with the sum σ, Hermite constructed a set of m + 1 polynomials
$${A_0}\left( z \right),{A_0}\left( z \right), \ldots ,{A_m}\left( z \right)$$
of degrees not exceeding a — σϱ 0, σϱ 1,..., σϱ m , respectively, such that all the functions
$${{\text{A}}_k}\left( z \right){e^{{\omega _l}z}} - {{\text{A}}_l}\left( z \right){e^{{\omega _k}z}}\,\left( {0 \leqq k < l \leqq m} \right)$$
vanish at z = 0 at least to the order σ + 1. On putting z = 1, these formulae produce simultaneous rational approximations of the numbers 1, e, e 2,..., e m that are so good that they imply the linear independence of these numbers and hence the transcendency of e.

Keywords

Positive Integer Linear Independence Common Multiple Integral Coefficient Large Positive Integer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Ch. Hermite 1873, Oeuvres, t. III, 151–181.Google Scholar
  2. Ch. Hermite 1893, Oeuvres, t. IV, 357–377.Google Scholar
  3. K. Mahler 1931, J. reine angew. Math., 166, 118–137.Google Scholar
  4. K. Mahler 1953, Phil. Trans. Royal Soc., A, 245, 371–398.MathSciNetMATHCrossRefGoogle Scholar
  5. K. Mahler 1953a, Proc. Acad. Amsterdam, A, 56, 30–42.MathSciNetGoogle Scholar
  6. Rosser and Schoenfeld 1962, Illinois J. of Math., 6, 64–94.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • K. Mahler
    • 1
  1. 1.Institute of Advanced StudiesAustralian National UniversityCanberraAustralia

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