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

Open Mathematics (2006)

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

## Access Full Article

top## Abstract

top## How to cite

topClaude 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

top- [1] H. Cartan: Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes, Hermann, Paris, 1961. Zbl0094.04401
- [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] 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] C. Gauthier: “Quelques propriétés algébriques des ensembles de nombres à inverse additif composé”, Ann. Sci. Math. Québec, Vol. 26, (2002), pp. 47–59.
- [5] I.J. Good: “A simple generalization of analytic function theory”, Expo. Math., Vol. 6 (1988), pp. 289–311. Zbl0662.30045
- [6] E. Grosswald: Topics from the Theory of Numbers, Birkhäuser, Boston, 1984.
- [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] H. Silverman: Complex Variables, Houghton Mifflin, Boston, 1975.
- [9] G. Valiron: Théorie des fonctions, Masson, Paris, 1948.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.