On the periodicity of trigonometric functions generalized to quotient rings of R[x]

Claude Gauthier

Open Mathematics (2006)

  • Volume: 4, Issue: 3, page 395-412
  • ISSN: 2391-5455

Abstract

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We apply a method of Euler to algebraic extensions of sets of numbers with compound additive inverse which can be seen as quotient rings of R[x]. This allows us to evaluate a generalization of Riemann’s zeta function in terms of the period of a function which generalizes the function sin z. It follows that the functions generalizing the trigonometric functions on these sets of numbers are not periodic.

How to cite

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Claude Gauthier. "On the periodicity of trigonometric functions generalized to quotient rings of R[x]." Open Mathematics 4.3 (2006): 395-412. <http://eudml.org/doc/269604>.

@article{ClaudeGauthier2006,
abstract = {We apply a method of Euler to algebraic extensions of sets of numbers with compound additive inverse which can be seen as quotient rings of R[x]. This allows us to evaluate a generalization of Riemann’s zeta function in terms of the period of a function which generalizes the function sin z. It follows that the functions generalizing the trigonometric functions on these sets of numbers are not periodic.},
author = {Claude Gauthier},
journal = {Open Mathematics},
keywords = {30G35; 11M41},
language = {eng},
number = {3},
pages = {395-412},
title = {On the periodicity of trigonometric functions generalized to quotient rings of R[x]},
url = {http://eudml.org/doc/269604},
volume = {4},
year = {2006},
}

TY - JOUR
AU - Claude Gauthier
TI - On the periodicity of trigonometric functions generalized to quotient rings of R[x]
JO - Open Mathematics
PY - 2006
VL - 4
IS - 3
SP - 395
EP - 412
AB - We apply a method of Euler to algebraic extensions of sets of numbers with compound additive inverse which can be seen as quotient rings of R[x]. This allows us to evaluate a generalization of Riemann’s zeta function in terms of the period of a function which generalizes the function sin z. It follows that the functions generalizing the trigonometric functions on these sets of numbers are not periodic.
LA - eng
KW - 30G35; 11M41
UR - http://eudml.org/doc/269604
ER -

References

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  1. [1] H. Cartan: Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes, Hermann, Paris, 1961. Zbl0094.04401
  2. [2] P. Deguire and C. Gauthier: “Sur la dérivation dans certains anneaux quotients de R[x]”, Ann. Sci. Math. Québec, Vol. 24, (2000), pp. 19–31. Zbl1094.13547
  3. [3] L. Euler: “De summis serierum reciprocarum”, Comment. Acad. Sci. Petropolit., Vol. 7(1734/35), (1740), pp. 123–134; Opera omnia, Ser. 1, Vol. 14, Leipzig-Berlin, 1924, pp. 73–86. 
  4. [4] C. Gauthier: “Quelques propriétés algébriques des ensembles de nombres à inverse additif composé”, Ann. Sci. Math. Québec, Vol. 26, (2002), pp. 47–59. 
  5. [5] I.J. Good: “A simple generalization of analytic function theory”, Expo. Math., Vol. 6 (1988), pp. 289–311. Zbl0662.30045
  6. [6] E. Grosswald: Topics from the Theory of Numbers, Birkhäuser, Boston, 1984. 
  7. [7] M.E. Muldoon and A.A. Ungar: “Beyond sin and cos”, Math. Mag., Vol. 69 (1996), pp. 2–14. http://dx.doi.org/10.2307/2691389 Zbl0860.33003
  8. [8] H. Silverman: Complex Variables, Houghton Mifflin, Boston, 1975. 
  9. [9] G. Valiron: Théorie des fonctions, Masson, Paris, 1948. 

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